Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Mathematics for AIEEE - CET 2009-10
Document related concepts
no text concepts found
Transcript
MATHEMATICS
Permutation & Combination
1.
If a and b are the greatest values of 2nCr and 2n−1Cr respectively. Then
(a) a = 2b
(b) b = 2a
(c) a = b
2.
If n is even and C0 < C1 < C2 < .... < nCr > nCr+1 > nCr+2 > .... > nCn, then r =
5.
6.
7.
8.
n
n
2
(b)
n −1
2
(c)
n−2
2
(d)
n+2
2
∑
k =m
4.
n
(a)
n
3.
(d) none of these
n
k
Cr equals
(a) n+1Cr+1
(b) n−1Cr+1 − mCr
(c) n+1Cr+1 − nCr+1
(d) n+1Cr+1 + mCr+1
The number of signals that can be generated by using 6 differently coloured
flags, when any number of them may be hoisted at a time is
(a) 1956
(b) 1957
(c) 1958
(d) 1959
The sum of all five digit numbers that can be formed using the digits 1, 2, 3, 4,
5, when reptition of digits is not allowed, is
(a) 366000
(b) 660000
(c) 360000
(d) 3999960
Three dice are rolled. The number of possible outcomes in which at least one
die shows 5 is
(a) 215
(b) 36
(c) 125
(d) 91
The number of times the digit 3 will be writen when listing the integers from 1
to 1000 is
(a) 269
(b) 300
(c) 271
(d) 302
The number of ways in which a mixed double game can be arranged from
amongst 9 married couples if no husband and wife play in the same game is
(a) 756
(b) 1512
(c) 3024
(d) none of these
1
Punjab EDUSAT Society (PES)
MATHEMATICS
9.
10.
11.
12.
13.
14.
15.
16.
17.
The number of different seven digit numbers that can be written using only the
three digit 1, 2 and 3 with the condition that the digit 2 occurs twice in each
number is
(a) 7P2 25
(b) 7C2 25
(c) 7C2 52
(d) none of these
In a certain test there are n questions. In this test 2n−k students gave wrong
answers to at least k questions, where k = 1, 2, 3, ..., n. If the total number of
wrong answers given is 2047, then n is equal to
(a) 10
(b) 11
(c) 12
(d) 13
The number of integers which lie between 1 and 106 and which have the sum
of digits is equal to 12 is
(a) 8550
(b) 5382
(c) 6062
(d) 8055
The number of integral solutions of x + y + z = 0, with x ≥ −5, y ≥ −5, z ≥ −5
is
(a) 135
(b) 136
(c) 455
(d) 105
The number of ways in which a score of 11 can e made from a throw by three
persons, each throwing a single die once, is
(a) 45
(b) 18
(c) 27
(d) none of these
If n = mC2, the value of nC2 is given by
(a) m+1C4
(b) m−1C4
(c) m+2C4
(d) none of these
The number of positive integers satisfying the inequality
≤ 100 is
(a) 9
(b) 8
(c) 5
(d) none of these
m+1
Cn−2 −
n+1
Cn−1
Out of 10 consonants four vowels, the number of words that can be formed
using six consonants and three vowels is
(a) 10P6 × 6P3
(b) 10C6 × 6C3
(c) 10C6 × 4C3 × 9!
(d) 10P6 × 4P3
If x, y, z, are (m + 1) distinct prime numbers, the number of factors of xnyz...is
(a) m(m + 1)
(b) 2nm
(c) (n + 1)2m
(d) n2m
2
Punjab EDUSAT Society (PES)
MATHEMATICS
18.
19.
20.
21.
22.
23.
24.
25.
26.
The number of ways in which one or more balls can be selected out of 10
white 9 green and 8 blue balls is
(a) 892
(b) 881
(c) 891
(d) 879
The number of all three elements subsets of the set {a1, a2, a3, ... an} which
contains a3 is
(a) nC3
(b) n−1C3
(c) n−1C2
(d) none of these
The number of non negative integral solutions of x + y + z ≤ n , where n ∈ N
is
(a) n+3C3
(b) n+4C4
(c) n+5C5
(d) none of these
If n objects are arranged in a row, then the number of ways of selecting three
of these objects so that no two of them are next to each other is
(a) n−2C3
(b) n−3C2
(c) n−3C3
(d) none of these
The number of ways in which an examiner can assign 30 marks to 8 questions,
giving not less than 2 marks to any question is
(a) 21C7
(b) 21C6
(c) 21C8
(d) none of these
If 56Pr+6 : 54Pr+3 = 30800 : 1, then the value of r is
(a) 40
(b) 41
(c) 42
(d) none of these
If n+2C8 : n−2P4 = 57 : 16, then the vale of n is
(a) 20
(b) 19
(c) 18
(d) 17
Ten different letter of an alphabet are given words with five letters are formed
from these given letters. Then the number of words which have at least one
letter repeated is
(a) 33
(b) 44
(c) 48
(d) 52
A box contains two white balls, three lack balls and four red balls. In how
many ways can three balls be drawn from the box if at least one black ball is
to be included in the draw
(a) 129
(b) 84
(c) 64
(d) none of these
3
Punjab EDUSAT Society (PES)
MATHEMATICS
27.
28.
29.
30.
31.
32.
33.
The letters of the word RANDOM are written in all possible orders and these
words are written out as in a dictionary then the rank of the word RANDOM is
(a) 614
(b) 615
(c) 613
(d) 616
If the letters of the word MOTHER are written in all possible orders and these
words are written out as in a dictionary, then the rank of the word MOTHER is
(a) 240
(b) 261
(c) 308
(d) 309
m men and n women are to be seated in a row so that no two women sit
together. If m > n, then the number of ways in which they can be seated is
m!(m + 1)!
(m − n + 1)!
(a)
n!n!
(m + n)!
(b)
(c)
m!n!
(m − n + 1)!
(d) none of these
A five digit number divisible by 3 is to be formed using numerals 0, 1, 2, 3, 4,
and 5 without repitition. The total number of ways this can be done
(a) 216
(b) 240
(c) 3125
(d) 600
A committee of 5 is to be formed from 9 ladies and 8 men. If the committee
commands a lady majority, then the number of ways this can be done
(a) 2352
(b) 1008
(c) 3360
(d) 3486
The number of ways in which 52 cards can be divided into 4 sets, three of
them having 17 cards each and fourth one having just one card
(a)
52!
(17!)3
(b)
52!
(17!)3 3!
(c)
51!
(17!)3
(d)
51!
(17!)3 3!
m parallel lines in a plane are intersected y a set of n parallel lines. The total
number of parallelograms so formed is
(a)
(m − 1)(n − 1)
4
(b)
mn
4
(c)
m(m − 1)n(n − 1)
2
(d)
mn(m − 1)(n − 1)
4
4
Punjab EDUSAT Society (PES)
MATHEMATICS
34.
35.
36.
37.
38.
39.
40.
There are n straight lines in a plane, no two of which are parallel, and no three
pass through the same point. Their points of intersection are joined. The
number of fresh lines thus obtained as
(a)
n(n − 1)(n − 2)
8
(b)
n(n − 1)(n − 2)(n − 3)
6
(c)
n(n − 1)(n − 2)(n − 3)
8
(d) none of these
The straight lines I1, I2, I3 are parallel and lie in the same plane. A total
number of m points are taken on I1; n points on I2;k points on I3. The
maximum number of triangles formed with vertices at these points are
(a) m+n+kC3
(b) m+n+kC3 − mC3 − nC3 − kC3
(c) mC3 + nC3 + kC3
(d) none of these
The number of ways in which we can distribute mn students equally among m
sections is given by
(a)
(mn)!
n!
(b)
(mn)!
(n!)m
(c)
(mn)!
m!n!
(d) (mn)m
If a polygon has 54 diagonals, the number of its sides is given by
(a) 12
(b) 11
(c) 10
(d) 9
The number of ways in which we can arrange 4 letters of the word
MATHEMATICS is given by
(a) 136
(b) 2454
(c) 1680
(d) 192
Each of the five questions in a multiple-choice test has 4 possible answers.
The number of different sets of possible answers is
(a) 45 − 4
(b) 54 − 5
(c) 1024
(d) 624
The number of squares which we can form on a chessboard is
(a) 64
(b) 160
(c) 224
(d) 204
5
Punjab EDUSAT Society (PES)
MATHEMATICS
Binomial Theorem
1.
If n > 3, then
xyz C0 + (x − 1) (y − 1) (z − 1) C1 + (x − 2) (y − 2) (z − 2) C2 + .... + ....... (x −
n) (y − n) (z − n) Cn equals
2.
(a) xyz
(b) nxyz
(c) −xyz
(d) none of these
The total number of dissimilar terms in the expansion of (x1 + x2 + .... + xn)3
is
(a) n3
(c)
3.
4.
n(n + 1)(n + 2)
6
(b)
n 3 + 3n 2
4
(d)
n 2 (n + 1)2
4
The coefficient of x6 in the expansion of (1 + x 2 − x 3 )8 is
(a) 80
(b) 84
(c) 88
(d) 92
The digit at units place in the number 171995 + 111995 − 71995 is
(a) 0
(b) 1
(c) 2
(d) 3
(−1)r
∑ n C equals
r =0
r
n
5.
If n is an odd natural number, then
(a) 0
(c)
6.
7.
8.
n
2n
(b) 1/n
(d) none of these
If the second, third, and fourth term in the expansion of (x + a)n are 240, 720,
1080 respectively, then the value of n is
(a) 15
(b) 20
(c) 10
(d) 5
If n ∈ N such that (7 + 4 3) n = 1 + F where I ∈ N and 0 < F < 1, then the
value of (1 + F) (1 − F) is
(a) an even integer
(b) an odd integer
(c) depends upon n
(d) none of these
The sum of coefficients of the polynomial (1 + x − 3x2)2143
(a) 0
(b) 1
(c) 72n
(d) 22n
6
Punjab EDUSAT Society (PES)
MATHEMATICS
9.
10.
11.
If n is an even positive integer, then the condition taht the greatest term in the
expansion of (1 + x)n may have the greatest coefficient also is
(a)
n
n+2
<x<
n+2
n
(b)
(c)
n
n+4
<x<
n+4
4
(d) none of these
If the rth, (r + 1)th and (r + 2)th coefficient of (1 + x)n are in AP, then n is a root
of the equation
(a) x2 − x(4r + 1) + 4r2 − 2 = 0
(b) x2 + x(4r + 1) + 4r2 − 2 = 0
(c) x2 + x (4r + 1) + 4r2 + 2 = 0
(d) none of these
If C0, C1, C2, ..., Cn are binomial coefficients in the expansion of (1 + x)n, then
C C C
C
the value of C0 − 1 + 2 − 3 ....(−1) n n
2
3
4
n +1
(a) 0
(c)
12.
n +1
n
<x<
n
n +1
(b)
2n
n +1
1
n +1
(d) −
1
n +1
If n is an integer greater than unity, they the value of
a − nC1(a − 1) + nC2 (a − 2) − nC2 (a − 3) + ..... + (−1)n (a − n) is
13.
(a) 0
(b) 1
(c) n
(d) −1
If Cr stands for nCr, then the sum of the series
n n
2 ! !
2
2
a − [C02 − 2C12 + 3C22 + .... + (−1) n (n + 1)Cn2 ]
n!
(a) 0
(c)
n
(−1) 2 (n + 2)
(b)
n
(−1) 2 (n + 1)
(d)
n
(−1) 2 n
10
14.
15.
x 2
If the rth term in the expansion of − 2
3 x
(a) 2
(b) 3
(c) 4
(d) 5
contains x4, then r is equal to
The value of C02 + 3C12 + 5C22 + ...... to (n + 1) terms, is
(a)
2n −1
C n −1
(b) (2n + 1) 2n−1Cn
7
Punjab EDUSAT Society (PES)
MATHEMATICS
(c) 2(n + 1) 2n-1Cn−1
16.
17.
(d) 2n−1Cn + (2n+1) 2n−1Cn−1
The number of terms in the expansion of (x + y + z)10 is
(a) 11
(b) 33
(c) 66
(d) 1000
If x is so small that its two and higher powers can be neglected and if
−
1
(1 − 2x) 2 (1 − 4x)
−
5
2
= 1 + kx , then k is equal to
(a) 1
(b) −2
(c) 10
(d) 11
10
18.
19.
20.
21.
b
The number of terms independent of x in 1 + x +
x
(a) 5
(b) 6
(c) 7
(d) 8
The coefficient of x8 in the expansion of (1 + x)10 (1 − 2x + x 2 ) is
(a) 10C6 + 10C7 + 10C8
(b) 10C6 − 10C7 + 10C8
(c) 10C6 − 210C7 + 10C8
(d) 10C6 + 210C7 + 10C8
The sum of the coefficient in the expansion of (1 − 2x − 3x2)20 is
(a) 240
(b) 320
(c) 18763
(d) 10840
The sum of the series C02 + C12 + C22 + .... + Cn2 is
(a)
2n (1.3.5.7...(2n − 1))
n!
(c) C(2n, n + 1)
22.
are
(b)
2n (2.4.6...2n)
n!
(d)
2n (2.4.6...2n)
n!
The number of zero at the end of 1271 is
(a) 29
(b) 30
(c) 31
(d) 32
8
Punjab EDUSAT Society (PES)
MATHEMATICS
SECTION - A
1.
Find the ratio in which the line segment joining the points (1, 2) and (−2, 1) is
divided by the line 3x + 4y = 7.
2.
A rod of length 1 slides with its ends on two perpendicular lines. Find the
locus of the mid-point.
3.
Prove that the line 5x − 2y − 1 = 0 is mid parallel to the lines 5x − 2y − 9 = 0
and
5x − 2y + 7 = 0.
4.
Show that the area of the triangle with vertices at (p − 4, p + 5), (p + 3, p − 2)
and
(p, p) is independent of p.
5.
Find the circumcentre of the triangle whose sides are 3x − y + 3 = 0, 3x + 4y +
3 = 0 and x + 3y + 11 = 0.
6.
A variable line passes through the point of intersection of the straight lines
x y
x y
+ = 1 and + = 1 cuts the coordinate axes its A and B respectively.
a b
b a
Find the locus of the mid-point AB
7.
Prove that the quadrilateral whose vertices are A(−2, 5), B(4, −1), C(9, 1) and
D(3, 7) is parallelogram, and find its area. Find the coordinates of a point E on
AC such that it divides AC in the ratio 2 : 1. Prove that D, E and F, the mid
point of BC are collinear.
8.
Find the line which is parallel to x-axis and crosses thr curve y = x at an
angle of 45o.
9.
Show that the straight lines 7x − 2y + 10 = 0, 7x + 2y − 10 = 0 and y + 2 = 0
form an isosceles triangle, and find its area.
10.
Vertices of a ∆ABC are A(2, 3), B(−4, −4), C(5, −8). Then find the length of
the median through C.
11.
If p be the length of perpendicular pass the origin on the line
x y
+ = 1 then
a b
1
1 1
= 2+ 2.
2
p
a
b
12.
If P(1, 0), Q(−1, 0) and R(2, 0) are three given points, then find the locus of
point S satisfying the relation SQ2 + SR2 = 2SP2.
13.
Determine all values of α for which the point (α, α2) lies inside the triangle
formed by lines 2x + 3y − 1 = 0, x + 2y − 3 = 0 and 5x − 6y − 1 = 0.
9
Punjab EDUSAT Society (PES)
MATHEMATICS
14.
If the line 3x + 4y − 24 = 0 meets the coordinate axes at A and B, then find the
incenter of the ∆OAB .
15.
The vertices of a triangle are A(−1, −7), B(5, 1) and C(1, 4). Find the equation
of the bisector of the angle ∠ ABC
16.
A line through A(−5,−4) meets the line x + 3y + 2 = 0,2x + y + 4 = 0 and x − y
2
2
2
15 10 6
−5 = 0 at the points B, C and D respectively. If
+
=
,
AB AC AD
find the equation of the line.
17.
18.
A rectangle PQRS has its side PQ parallel to the line y = mx and vertices P, Q,
and S on the lines y = a, x = b and x = −b respectively. Find the locus of vertex
R.
1
Let O(0, 0), A (2, 0) and B 1,
be the vertices of a triangle. Let R be the
3
region consisting of all those points P inside ∆OAB which satisfy
d(P, OA) ≤ min [d (P, OB), d(P, AB)]
where d denotes the distance from the point to the corresponding line. Sketch
the region R and find its area.
19.
Using co-ordinate geometry, prove that the three altitudes of any triangle are
concurrent.
20.
Show that the centroid ‘G’ of a triangle divides the join of its orthocentre H
and circumcentre S in the ratio 2 : 1.
10
Punjab EDUSAT Society (PES)
MATHEMATICS
SECTION - B
1.
2.
3.
4.
5.
6.
7.
Find the ratio in which the line segment joining the points (1, 2) and (−2, 1) is
divided by the line 3x + 4y = 7.
(a) (3, 7)
(b) (2, 4)
(c) (7, −2)
(d) none of these
The triangle joining the points (2, 7), (4, −1), (2, 6) is
(a) equilateral
(b) right angled
(c) Isoceles
(d) scalene
The co-ordinates of the foot of the perpendicular from the point (2, 4) on the
line x + y = 1 are
1 3
(a) ,
2 2
−1 3
(b) ,
2 2
4 1
(c) ,
3 2
3 −1
(d) ,
4 2
The points (2a, a), (a, 2a) and (a, a) enclose a triangle of area 8 units if
(a) a = −4
(b) a = 4
(c) a = 2 2
(d) none of these
The straight lines x + y = 0, 3x + y − 4 = 0, x + 3y − 4 = 0 from a triangle
which is
(a) equilateral
(b) right angled
(c) isoceles
(d) none of these
Every line of the system (1 + 2λ)x + (λ − 1) y + 3 = 0, λ being a parameter
passes through a fixed point A. The equation of the line through A and parallel
to
the
line
3x − y = 0 is
(a) 3x − y + 5 = 0
(b) −3x + y + 5 = 0
(c) 3x − y + 6 = 0
(d) 3x − y + 8 = 0
The lines 3x2 − y2 = 0 and x = 4 enclose a triangle which is
(a) right angled
(b) equilateral
(c) isoceles
(d) none of these
11
Punjab EDUSAT Society (PES)
MATHEMATICS
8.
9.
10.
11.
12.
13.
14.
If the lines x + ay + a = 0, bx + y + b = 0 and cx + cy + 1 = 0 (a, b, c being
a
b
c
distinct ≠ 1) are concurrent, the value of
is
+
+
a −1 b −1 c −1
(a) 1
(b) 7
(c) 10
(d) 13
If the distance of any point (x, y) from the origin is defined as (x, y) = max
{|x|, |y|} d(x, y) = a non-zero constant, then locus is
(a) circle
(b) straight line
(c) square
(d) none of these
The reflection of the point (4, −13) in the line 5x + y + 6 = 0 is
(a) (−1, −14)
(b) (3, 4)
(c) (1, 2)
(d) (−4, 15)
The acute angle θ through which the coordinate axis should be rotated for the
point A(2, 4) to attain the new abscissa 4 is given by
(a) tan θ =
3
4
(b) tan θ =
5
6
(c) tan θ =
7
8
(d) none of these
If one of the diagonal of the square is along x = 2y and one of its vertices is (3,
0), then its, sides through the vertex is given by the equations
(a) y − 3x + 3 = 0, 3y + x + 9 = 0
(b) y − 3x + 9 = 0, 3y − x + 3 = 0
(c) y − 3x + 9 = 0, 3y + x −3 = 0
(d) x + 3x − 3 = 0, 3y + x + 9 = 0
The bisector of the acute angle formed between the lines 4x − 3y + 7 = 0 and
3x − 4y + 14 = 0 has the equations
(a) x + y − 7 = 0
(b) x − y + 3 = 0
(c) 3x + y − 11 = 0
(d) none of these
Let the equation y − y1 = m(x − x1). If m and x1 are fixed and different lines
are drawn for different values of y, then
(a) the lines will p on through the fixed point
(b) there will be a set of parallel lines
(c) all the lines will intersect the line x = x1
(d) none of these
12
Punjab EDUSAT Society (PES)
MATHEMATICS
15.
The coordinates of the point P on the line 2x + 3y + 1 = 0 such that |PA − PB|
is maximum where A is (2, 0) and B is (0, 2) is
(a) (7, −5)
(b) (7, 6)
(c) (0, −5)
(d) (0, 5)
SECTION - C
1.
2.
3.
4.
5.
The orthocenter of the triangle formed by the line x + y = 1, 2x + 3y = 6 and
4x − y + 4 = 0 lies in
(a) I Quad.
(b) II Quad
(c) III Quad.
(d) IV Quad
The image of the point A(1, 2) be the line mirror y = x is the point B and
image of B by the line mirror y = 0 is the point (α, β) then
(a) α = 1, β = −2
(b) α = 0, β = 0
(c) α = 2, β = −1
(d) α = 2, β = 2
The equation of the line which passes through (acos3θ, a sin3θ) and
perpendicular to the line x secθ + y cosecθ = a is
(a) x sinθ + y cosθ = 2a cos2θ
(b) x cosθ + y sinθ = 2a cos2θ
(c) x sinθ + y cosθ = 2 sin2θ
(d) x cosθ − y sinθ = a cos2θ
The equation of the straight line which passes through the point (1, −2) and
cuts
x equal intercepts from axes will be
(a) x + y = 1
(b) x − y = 1
(c) x + y + 1 = 0
(d) none of these
The orthocentre of the triangle whose vertices are (0, 0), (3, 0) and (0, 4) is
3
(a) ,
2
2
(b) (0, 0)
13
Punjab EDUSAT Society (PES)
MATHEMATICS
4
(c) 7,
3
6.
7.
8.
9.
10.
11.
The distance between the lines 4x + 3y = 11 and 8x + 6y = 15 is
(a) 4
(b) 7/2
(c) 4/10
(d) 7/15
A point equidistant from the lines 4x + 3y + 10 = 0, 5x − 12y + 26 = 0 and
7x + 24y − 50 = 0 is
(a) (0, 0)
(b) (0, 1)
(c) (1, −1)
(d) (1, 1)
The lines 2x + y − 1 = 0, ax + 3y − 3 = 0 and 3x + 2y − 2 = 0 are concurrent
for
(a) a = 0 only
(b) all value of a
(c) −1 ≤ a ≤ 3
(d) a > 0 only
The straight line passes through the point (1, 1) and the portion of the lines
intercepted between x and y axis is divided at the point in the ratio 3 :4. The
equation of the line is
(a) 4x + 3y = 7
(b) 3x + 4y = 7
(c) 2x − 3y = 1
(d) 3x − 4y + 1 = 0
Area of the rhombus enclosed by the lines ax ± by ± c = 0 is
(a) 2c2/ab
(b) 2b2/ca
(c) 2a2/bc
(d) none of these
If a line making an angle θ with the positive direction of x axis, and is drawn
through the point (2, 1) to intersect the line y − 2x + 6 = 0 at a distance 3 2
from the point, then tanθ is equal to
1
7
(b) 1
(c) 8
(d) 7
(a)
12.
(d) (4, −4)
If A and B are two points having coordinate (3, 4) and (5, −2) respectively and
P is a point such that PA = PB and area of triangle PAB = 10 sq.units then
coordinates of P are
(a) (7, 2) or (1, 0)
(b) (7, 4) or (13, 2)
(c) (4, 13) or (2, 2)
(d) none of these
14
Punjab EDUSAT Society (PES)
MATHEMATICS
13.
The equation of the locus of the point which moves so that the sum of the
distance from two given points (ae, 0) and (−ae, 0) is equal to 2a, is
(a)
y2 x 2
−
=1
b2 a 2
(b)
x 2 y2
(c) 2 + 2 = 1
b
a
14.
15.
x 2 y2
−
=1
a 2 b2
x 2 y2
(d) 2 + 2 = 1
a
b
The incentre of the triangle formed by the lines x = 0, y = 0 and 3x + 4y = 12
is
(a) (1, 1)
1 1
(b) ,
2 2
1
(c) , 1
2
1
(d) 1,
2
The value of λ for which the following three lines x + y − 1 = 0, λx + 2y − 3 =
0,
λ2x + 4y + 9 = 0 are concurrent are
(a) 2, −15
(b) 3, 14
(c) 2, 15
(d) 6, 17
ANSWER
SECTION – A
1.
4:9
2.
4x2 + 4y2 = 1
15
Punjab EDUSAT Society (PES)
MATHEMATICS
5.
3, −3
6.
ab(x + y) = 2xy(a + b)
8.
x=
9.
14 sq. units
10.
1
2
85
12.
Straight lie parallel to y-axis
13.
−3
< α < −1
2
14.
(2, 2)
15.
7y = x + 2
16.
2x + 3y + 22 = 0
17.
(m2 − 1)x − my + b(m2 + 1) + am = 0
18.
(2 − 3) sq.units
OBJECTIVE ANSWER
SECTION – A
1. (c)
(c)
2. (b)
3. (b)
4. (a)
5.
6. (a)
(a)
7. (c)
8. (a)
9. (b)
10.
11. (a)
(a)
12. (c)
13. (b)
14. (b, c)
15.
SECTION – B
1. (a)
(b)
2. (c)
3. (d)
4. (c)
5.
6. (c)
(a)
7. (a)
8. (b)
9. (a)
10.
11. (b)
(a)
12. (a)
13. (d)
14. (a)
15.
16
Punjab EDUSAT Society (PES)
MATHEMATICS
CIRCLES
If y = 3x + c is a tangent to the circle x2 + y2 – 2x – 4y – 5 = 0, then c is
(a)
8
(b)
9
(c)
10
(d)
5
2.
The points (-5, 11), (11, 19), (18, -4) lie in a circle. Centre of the circle is at
(a)
(3, 4) (b)
(4, 3) (c)
(5, 4) (d)
none of these
2
2
2
2
3.
The two circles x + y – 2x + 6y + 6 = 0 and x + y – 5x + 6y + 15 = 0 touch
each other.
The equation of their common tangent is
(a)
x = 3 (b)
y = 6 (c)
7x – 12y – 21 = 0
(d)
7x + 12y +
21 = 0
1.
4.
6 is
The length of the tangent from the point (5, 4) to the circle x2 + y2 + 2x – 6y =
21 (b)
38 (c)
2 2 (d)
2 13
(a)
5.
The area of circle centred at (1, 2) and passing through (4, 6) is
(a)
5π
(b)
10π (c)
25π (d)
none of these
6.
The equation of the circle passing through the intersection of x2 + y2 + 13x –
13y = 0 and
2x2 + 2y2 + 4x – 7y – 25 = 0 and whose centre lies on 13x + 30y = 0
(a)
x2 + y2 – 13x – 13y – 25 = 0
(b)
4x2 + 4y2 + 30x – 13y –
25 = 0
(d)
x2 + y2 + 30x –
(c)
2x2 + 2y2 + 30x – 13y – 25 = 0
13y + 25 = 0
7.
The locus of the centre of a circle radius 2 which rolls on the out side the
circle
x2 + y2 + 3x – 6y – 9 = 0 is
(a)
x2 + y2 + 3x – 6y + 5 = 0
(b)
x2 + y2 + 3x – 6y + 31 = 0
2
2
(c)
x + y + 3x – 6y + 29/4 = 0 (d)
none of these
8.
The distance between the chords of contact of the tangent to the circle x2 + y2
+ 2gx + 2fy + c = 0 from the origin and the point (g, f) is
(a)
g2 + f2
(b)
(c)
1 g 2 + f 2 + c2
2
g2 + f 2
(d)
(
1 2
g + f 2 + c2
2
1 g2 + f 2 − c
2 g2 + f 2
)
9..
The equation of circle with origin as centre and passing through the vertices of
an equilateral
triangle whose median is of length 3a is
(a)
x2 + y2 = 9a2
(b)
x2 + y2 = 16a2
2
2
2
(c)
x + y = 4a
(d)
x2 + y2 = a2
10.
A circle having area = 154 sq. units has two diameters 2x – 3y – 5 = 0 and 3x
– 4y – 7 = 0, then
the equation of the circle is
17
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
x2 + y2 – 2x + 2y – 62 = 0
(b)
x2 + y2 – 2x + 2y –
(c)
x2 + y2 + 2x – 2y – 47 = 0
(d)
x2 + y2 + 2x – 2y –
47 = 0
62 = 0
11.
=0
The number of common tangents to the circle x2 + y2 = 1 and x2 + y2 – 4x + 3
(a)
1
(b)
2
(c)
3
(d)
4
12. If the circles x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0 intersect
orthogonally then k equals
(a)
2 or -3/2
(b)
-2 or -3/2
(c)
2 or 3/2
(d)
-2 or 3/2
13.
Four distinct points (2k, 3k) (1, 0)(0, 1) and (0, 0) lie on a circle for
(a)
for all integral k
(b)
k < 0 (c)
0<k<1
(d)
for
two values of k
The equation of the circle concentric to the circle 2x2 + 2y2 – 3x + 6y + 2 = 0
14.
and having area
double the area of this circle is:
(a)
8x2 + 8y2 – 24x + 48y – 13 = 0
(b)
16x2 + 16y2 + 24x
– 48y – 13 = 0
(c)
16x2 + 16y2 – 24x + 48y – 13 = 0
(d)
8x2 + 8y2 + 24x –
48y – 13 = 0
15.
One of the diameter of the circle x2 + y2 – 12x + 4y + 6 = 0 is given by
(a)
x+y=0
(b)
x + 3y = 0
(c)
x = y (d)
3x + 2y =
0
16.
A variable chord is drawn through the origin to the circle x2 + y2 – 2ax = 0.
The locus of the
centre of the circle drawn on this chord as diameter is:
2
2
(a)
x + y + ax = 0
(b)
x2 + y2 – ax = 0
(c)
x2 + y2 +
2
2
ay = 0 (d)
x + y – ay = 0
17.
The value of k so that x2 + y2 + kx + 4y + 2 = 0 and 2(x2 + y2) – 4x – 3y + k =
0 cut orthogonally, is
(a)
10/3
(b)
- 8/3
(c)
- 10/3
(d)
8/3
2
2
18.
The length of the common chord of the circles x + y + 2x + 3y +1=0 and x2
+y2 +4x+ 3y+2 = 0 is
(a)
9/2
(b)
2 2
(c)
(d)
3 2
3
2
19. The limiting points of the system of circles represented by the equation
(
)
9
= 0 , are
2
3
± ,0 (b)
2
2 x2 + y 2 + λ x +
(a)
( 0,0 ) and
9
,0
2
9
(c) ± ,0
2
(d)
( ±3,0 )
18
Punjab EDUSAT Society (PES)
MATHEMATICS
20.
For the given circles x 2 + y 2 − 6 x − 2 y + 1 = 0 and x 2 + y 2 + 2 x − 8 y + 13 = 0
which of the following is true ?
(a)
One circle lies inside the other
(b)
One circle lies completely
outside the other
(c)
Two circles intersect in two points (d)
They touch each other
externally.
21.
If the chord y = mx + 1 subtends an angle of measure 45° at the major segment
of the circle
x2 + y2 = 1 then value of m is
1 ± 2 (b)
−2 ± 2
(c)
−1 ± 2
(d)
±1
(a)
22.
A variable circle passes through the fixed point A(p, q) and touches x-axis.
The locus of the other
end of the diameter through A is
(a)
(y – q)2 = 4px
(b)
(y – q)2 = 4py
2
(c)
(y – p) = 4qx
(d)
(y – p)2 = 4qy
23.
If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie along diameters of a circle of
circumference
10π, then the equation of the circle is
(a)
x2 + y2 + 2x – 2y – 23 = 0
(b)
x2 + y2 – 2x – 2y –
23 = 0
(c)
x2 + y2 + 2x + 2y – 23 = 0
(d)
x2 + y2 – 2x + 2y –
23 = 0
24.
The intercept on the line y = x by the circle x2 + y2 – 2x = 0 is AB. Equation
of the circle on AB as a diameter is
(a)
x2 + y2 + x – y = 0
(b)
x2 + y2 – x + y = 0
2
2
(c)
x +y +x+y=0
(d)
x2 + y2 – x – y = 0
25.
If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 – 3ax + dy – 1 = 0 intersect
in two distinct points P and Q then the line 5x + by – a = 0 passes through P and Q
for
(a)
exactly one value of a
(b)
no value of a
(c)
infinitely many values of a
(d)
exactly two values of a
26.
A circle touches the x-axis and also touches the circle with center at (0, 3) and
radius 2. The locus of the center of the circle is
(a)
an ellipse
(b)
a circle (c)
a hyperbola (d)
a parabola.
27.
If circle passes through the point (a, b) and cuts the circle x2 + y2 = k2
orthogonally, then the
equation of the locus of its center is
(a)
x2 + y2 – 3ax – 4by + (a2 + b2 – k2) = 0
(b)
2ax + 2by – (a2 –
b2 + k2) = 0
(c)
x2 + y2 – 2ax – 3by + (a2 – b2 – k2) = 0
(d)
2ax + 2by – (a2 +
2
2
b +k )=0
28.
If the pair of the lines ax2 + 2(a + b)xy + by2 = 0 lie along diameters of a circle
and divide the circle into four sectors such that the area of one of these sectors is
thrice the area of another
sector then
(a)
3a2 – 10ab + 3b2 = 0
(b)
3a2 – 2ab + 3b2 = 0
19
Punjab EDUSAT Society (PES)
MATHEMATICS
(c)
3a2 + 10ab + 3b2 = 0
(d)
3a2 + 2ab + 3b2 = 0
29.
If the lines 3x – 4y – 7 = 0 and 2x – 3y – 5 = 0 are two diameters of a circle of
area 49π square
units, the equation of the circle is
(a)
x2 + y2 – 2x + 2y – 62 = 0
(b)
x2 + y2 – 2x + 2y –
47 = 0
(c)
x2 + y2 + 2x – 2y – 47 = 0
(d)
x2 + y2 + 2x – 2y –
62 = 0
30.
Let C be the circle with centre (0, 0) and radius 3 units. The equation of the
2π
locus of the mid
points of the chords of the circle C that subtend an angle of
3
at its centre is
(a)
(c)
27
4
3
x2 + y 2 =
2
x2 + y 2 =
9
4
(b)
x2 + y 2 =
(d)
x2 + y 2 = 1
31.
Consider a family of circles which are passing through the point (-1, 1) and are
tangent to x- axis. If (h, k) are the co-ordinates of the centre of the circles,
then the
set of values of k is given by the interval.
(a)
k ≥ ½ (b)
-1/2 ≤ k ≤ ½ (c)
k≤½
(d)
0<k<½
32.
The point diametrically opposite to the point P(1, 0)on the circle
x 2 + y 2 + 2 x + 4 y − 3 = 0 is
(a)
( - 3, - 4)
(b)
( 3, 4)
(c)
( 3, -4)
(d)
( -3, 4)
2
2
33.
A square is inscribed in the circle x + y – 2x + 4y + 3 = 0. Its sides are
parallel to the co-ordinate
axes, then one vertex of the square is
1 + 2, −2
1 − 2, −2
1, −2 + 2
(a)
(b)
(c)
(d)
(
)
(
)
(
)
none of these
34.
Two circles x2 + y2 = 6 and x2 + y2 – 6x + 8 = 0 are given. Then the equation
of the circle through their points of intersection and the point (1, 1) is
(a)
x2 + y2 – 6x + 4 = 0
(b)
x2 + y2 – 3x + 1 = 0
(c)
none of these
(c)
x2 + y2 – 4y + 2 = 0
35.
If A and B are points in the plane such that
PA
= k (constant) for all P on a
PB
given circle, then we must have
(a)
k ∈ R – {0, 1} (b)
k ∈ R – {1} (c)
k ∈ R – {0} (d)
none of these
36.
If one of the diameters of the circle x2 + y2 – 2x – 6y + 6 = 0 is a chord to the
circle with centre
(2, 1), then the radius of the circle is
(a)
3
(b)
2
(c)
3
(d)
2
20
Punjab EDUSAT Society (PES)
MATHEMATICS
37.
A circle is given by x2 + (y – 1)2 = 1, another circle C touches it externally and
also the x-axis, then the locus of its centre is
(a)
{(x, y):x2 = 4y} ∪ {(x, y):y ≤ 0}
(b)
{(x, y):x2 + (y –
1)2 = 4} ∪ {(x, y):y ≤ 0}
(c)
{(x, y):x2 = y} ∪ {(0, y):y ≤ 0}
(d)
{(x, y):x2 = 4y} ∪
{(0, y):y ≤ 0 }
38.
Let ABCD be a quadrilateral with area 18, with side AB parallel to the side
CD
and AB = 2CD.
Let AD be perpendicular to AB and CD. If a circle is
drawn inside the quadrilateral ABCD
touching all the sides, then its radius is
(a)
3
(b)
2
(c)
3/2
(d)
1
ANSWERS(Circle )
1.
b
2.
d
3.
a
4.
a
5.
c
6.
a
7.
b
8.
d
9.
c
20.
b
11.
c
12.
a
13.
d
14.
c
15.
b
16.
b
17.
c
18.
b
19.
a
20.
d
21.
d
22.
d
23.
d
24.
d
25.
b
26.
d
27.
d
28.
d
29.
b
30.
b
31.
37.
c
d
32.
38.
a
b
33.
d
34.
b
35.
a
36.
c
21
Punjab EDUSAT Society (PES)
MATHEMATICS
1.
CONICS
The line y = mx + 1 is a tangent to the parabola y2 = 4x if
(a)
m=1
(b)
m=2
(c)
m=4
(d)
m=
3
2.
The point of intersection of the tangents at the ends of the latus rectum of the
parabola y2 = 4x is
(a)
(-1, 0)
(b)
(-1, -1)
(c)
(0, -1)
(d)
none of these
3.
Let E be the ellipse
x2 y 2
2
2
+
= 1 and C be the circle x + y = 4. Let P and Q be
9
4
the points (1, 2)
and (2, 1) respectively. Then
(a)
Q lies inside C but outside E
(b)
E
(c)
P lies inside both C and E
(d)
E
Q lies outside both C and
P lies inside C but outside
4.
The foci of the ellipse 25(x + 1)2 + 9(y + 2)2 = 225, are at
(a)
(-1, -2) and (-1, -6)
(b)
(-2, 1) and (-2, 6)
(c)
(-1, 2) and (-1, 6)
(d)
(-1, -20 and (-2, -1)
5.
The vertex of the parabola y2 = 4(a′ - a)(x – a) is
(a)
(a′, a) (b)
(a, a′) (c)
(a, 0) (d)
6.
is
(a′, 0)
If the line y = 3x + λ touches the hyperbola 9x2 – 5y2 = 45, then the value of λ
(a)
36
(b)
45
(c)
6
(d)
15
7.
If the parabola y2 = 4ax passes through (3, 2) then the length of its latusrectum is
(a)
2/3
(b)
4/3
(c)
1/3
(d)
4
8.
The eccentricity of the ellipse 9x2 + 5y2 – 30y = 0 is
(a)
1/3
(b)
2/3
(c)
¾
(d)
none of these
9.
the vertex of the parabola x2 + 8x + 12y – 8 = 0 is
(a)
(-4, 2) (b)
(4, -1) (c)
(-4, -1) (d)
(4, 1)
10.
P is any point on the ellipse 81x2 + 144y2 = 1944 whose foci are S and S′.
Then SP + S′P equals
4 6 (c)
36
(d)
324
(a)
3
(b)
11.
?
What is the equation of the central ellipse with foci (± 2, 0) and eccentricity ½
22
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
2
+ 3y = 12
3x2 + 4y2 = 48 (b)
4x2 + 3y2 = 48 (c)
x2 – 4y2 – 2x + 16y – 24 = 0 represents
(a)
straight lines (b)
an ellipse
parabola
3x2 + 4y2 = 12 (d)
4x2
a hyperbola
a
12.
13.
14.
(c)
The line lx +my + n = 0 is a normal to the ellipse
(
)
(
)
(a)
a 2 − b2
a 2 b2
+
=
m2 l 2
n2
(c)
a 2 − b2
a 2 b2
−
=
l 2 m2
n2
x2 y2
+
= 1 , if
a 2 b2
(
(b)
a2 − b2
a2 b2
+
=
l 2 m2
n2
(d)
none of these
If the straight line y = 2x + c is a tangent to the ellipse
be equal to
(a)
15.
(a)
(d)
)
x2 y 2
+
= 1 , then c will
8
4
±4
(b)
±6
(c)
±1
(d)
±8
The curve represented x = 3(cos t + sin t), y = 4(cos t – sin t) is
ellipse (b)
parabola
(c)
hyperbola
(d)
circle
16.
If the segment intercepted by the parabola y2 = 4ax with the line lx + my + n =
0 subtends a right
angle at the vertex, then
(a)
4al + n = 0
(b)
4al + 4am + n = 0
(c)
4am + n = 0 (d)
al + n = 0
17.
If the normal at any point P on the ellipse
g respectively,
(a)
a:b
18.
19.
then PG : Pg
(b)
a2 : b2 (c)
x2 y2
+
= 1 , meets the aces in G and
a 2 b2
b2 : a2
(d)
The curve given by x = cos 2t, y = sin t
(a)
ellipse (b)
circle (c)
part of parabola
Length of major axis of ellipse 9x2 + 7y2 = 63 is
(a)
3
(b)
9
(c)
6
(d)
b:a
(d)
hyperbola
2 7
20.
The angle between the pair of tangents drawn to the ellipse 3x2 + 2y2 = 5 from
the point (1, 2) is
23
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
(
12
tan −1
5
tan −1 12 5
(b)
(
tan −1 6 5
)
(c)
)
12
tan −1
5
(d)
21.
The number of parabolas that can be drawn if two ends of the latus rectum are
given, is
(a)
1
(b)
2
(c)
4
(d)
5
22.
Equation of tangent to hyperbola
x2 y 2
−
= 1 equally inclined to coordinate
3
2
axis is
(a)
y=x+1
(b)
y=x–1
(c)
y=x+2
(d)
y=
x–2
23.
The locus of a point whose distance from a fixed point and the fixed straight
line x = 9/2 is always in the ratio 2/3 is
(a)
hyperbola
(b)
ellipse
(c)
parabola
(d)
circle
24.
If e and e′ are the eccentricities of hyperbolas
hyperbola, then the
(a)
0
value of
(b)
x2 y 2
−
= 1 and its conjugate
a2 b2
1
1
+ 2 is:
2
e
e′
1
(c)
2
(d)
none of these
25.
In an ellipse the angle between the lines joining the foci with the positive end
of minor axis is a
right angle, the eccentricity of the ellipse is:
(a)
26.
(b)
1
3
(c)
The equation of tangent to the curve
(a)
1–x
27.
1
2
y=x±1
(b)
(d)
2
3
x2 y 2
−
= 1 which is parallel to y = x, is
3
2
y=x–½
(c)
y=x+½
(d)
y=
(d)
(0,
The foci of the conic section 25x2 + 16y2 – 150x = 175 are
(a)
(0, ± 3)
(b)
(0, ± 2)
(c)
(3, ± 3)
± 1)
28.
The product of perpendiculars drawn from any point of a hyperbola to its
asymptotes is
24
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
a 2b 2
a 2 + b2
(b)
a 2 + b2
a 2b 2
(c)
ab
(d)
a+ b
ab
a + b2
2
29.
The equation of the ellipse whose distance between the foci is equal to 8 and
distance between
the directrix is 18, is
2
(a)
5 x − 9 y 2 = 180 (b) 9 x 2 + 5 y 2 = 180 (c) x 2 + 9 y 2 = 180 (d)
5 x 2 + 9 y 2 = 180
30.
The eccentricity of the hyperbola can never be equal to
9
1
1
(a)
(b) 2
(c)
3
(d)
2
5
9
8
31.
If the line y = 2 x + λ be a tangent to the hyperbola 36 x 2 − 25 y 2 = 3600, then λ
is equal
(a)
16
(b)
-16
(c)
± 16 (d)
None of these
32.
The equation y 2 − x 2 + 2 x − 1 = 0 represents
(a)
a hyperbola
(b)
an ellipse
(c)
a pair of straight lines
(d)
a rectangular hyperbola
Two common tangents to the circle x2 + y2 = a2 and parabola y2 = 8ax are
(a)
x = ±(y + 2a) (b)
y = ±(x + 2a) (c)
x = ±(y + a) (d)
y=
±(x + a)
33.
34. The normal at ( bt12 , 2bt1 ) on parabola y2 = 4bx meets the parabola again in point
( bt
2
2 , 2bt 2
) , then
(a)
t2 = t1 +
t2 = t1 −
2
t1
2
t1
35. The foci of the ellipse
(b)
t2 = −t1 −
2
t1
(c)
t2 = −t1 +
2
t1
(d)
x2 y 2
x2
y2
1
+ 2 = 1 and the hyperbola
coincide. Then
−
=
16 b
144 81 25
the value of b2 is
(a)
9
(b)
1
(c)
5
(d)
7
36.
If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of
intersection of the parabola y2 = 4ax and x2 = 4ay, then
(a)
d2 + (3b – 2c)2 = 0
(b)
d2 + (3b + 2c)2 = 0
(c)
d2 + (2b – 3c)2 = 0
(d)
d2 + (2b + 3c)2 = 0
37.
The eccentricity of an ellipse, with its centre at the origin, is ½. If one of the
directrices is x = 4, then the equation of the ellipse is
(a)
4x2 + 3y2 = 1 (b)
3x2 + 3y2 = 12 (c)
4x2 + 3y2 = 12 (d)
3x2
2
+ 3y = 1
38.
Let P be the point (1, 0) and Q a point on the locus y2 = 8x. The locus of mid
point of PQ is
25
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
y2 – 4x + 2 = 0 (b)
x2 + 4y + 2 = 0
39.
(c)
x2 – 4x + 2 = 0 (d)
The locus of a point P(α, β) moving under the condition that the line y = αx +
β is a tangent to
(a)
hyperbola
40.
y2 + 4x + 2 = 0
the hyperbola
an ellipse
(b)
x2 y 2
−
= 1 is
a2 b2
a circle (c)
a parabola
(d)
a
a2 x2 a2 x
+
− 2a is
3
2
105
3
xy =
(d)
xy =
64
4
The locus of the vertices of the family of parabolas y =
(a)
xy =
35
16
(b)
xy =
64
105
(c)
x2
y2
For the Hyperbola
−
= 1, which of the following remains
41.
cos 2 α sin 2 α
constant
when α varies ?
(a)
Directrix
(b) Abscissae of vertices (c) Abscissae of foci (d)
Eccentricity
42.
The equation of a tangent to the parabola y 2 = 8 x is y = x + 2 .The point on
this line
from which the other tangent to the parabola is perpendicular to the given
tangent is
(a)
(0, 2)
(b) (2, 4)
(c)
(−2,0)
(d)
(−1,1)
43*. The normal to a curve at P(x ,y) meets the x-axis at G .If the distance of G
from the origin is twice the abscissa of P, then the curve is a
(a)
parabola
(b)
circle (c)
hyperbola
(d)
ellipse
A focus of an ellipse is at the origin. The directrix is the line x = 4 and the
1
eccentricity is . Then
the length of the semi-major axis is
2
4
5
8
2
(a)
(b)
(c)
(d)
3
3
3
3
45.
A parabola has the origin as its focus and the line x = 2 as the directrix . Then
the vertex of the
parabola is at
(a)
(0,1)
(b)
(2, 0)
(c)
(0, 2)
(d) (1, 0)
A hyperbola, having the transverse axis of length 2sin θ, is confocal with the
46.
ellipse
3x2 + 4y2 =12. Then its equation is
44.
(a)
(c)
x 2 cos ec 2 θ − y 2 sec 2 θ = 1
x 2 sin 2 θ − y 2 cos 2 θ = 1
(b)
(d)
x 2 sec 2 θ − y 2 cos ec 2 θ = 1
x 2 cos 2 θ − y 2 sin 2 θ = 1
26
Punjab EDUSAT Society (PES)
MATHEMATICS
ANSWERS(Conics)
1.
a
2.
a
3.
d
4.
c
5.
c
6.
c
7.
b
8.
b
9.
a
10.
b
11.
a
12.
c
13.
b
14.
b
15.
a
16.
a
17.
c
18.
c
19.
c
20.
c
21.
b
22.
a
23.
b
24.
b
25.
a
26.
a
27.
c
28.
a
29.
d
30.
b
31.
c
32.
c
33.
b
34.
b
35.
d
36.
d
37.
b
38.
a
39.
d
40.
c
41.
c
42.
c
43.
c
44.
c
45.
d
46.
a
27
Punjab EDUSAT Society (PES)
MATHEMATICS
Function
1.
3 is
The function f: N → N (N is the set of natural numbers) defined by f(n) = 2n +
(a)
(c)
2.
surjective
not injective
(b)
(D)
not surjective
none of these
If e x = y + 1 + y 2 , then y is equal to
(a)
e x + e− x
(b)
e x − e− x
(c)
e x − e− x
2
(d)
e x + e− x
2
The equation sin x - π/2 + 1 = 0 has one root in interval
π
π
3π
(b)
(c)
(a)
0,
,π
π,
2
2
2
of these
3.**
4.
3x + x3
1+ x
If f(x) = log
and
g
x
=
, then f[g(x)] equals
(
)
1 + 3x 2
1− x
(a)
-f(x)
(b)
3f(x)
(c)
(f(x))3
(d) none
(d)
-
3f(x)
5x − x2
The domain of the function f ( x ) = log10
is
4
(a)
[1, 4] (b)
(0, 5) (c)
(1, 5) (d)
(0, 4)
π π
6.. The largest interval lying in − , for which f(x) = 4 - x2 +
2 2
x
cos −1 − 1 + log ( cos x ) is defined is
2
π π
π
(a)
(b)
[0, π]
(c)
0,
− 4 , 4
2
π π
− 2 , 2
5.
7.
The value of n ∈ I, for which the period of the function f ( x ) =
(a)
8.
-3
(b)
3
(c)
2
(d)
1+ x
The function f ( x ) = log
satisfy the equation
1− x
(a)
f ( x1 ) f ( x2 ) = f ( x1 + x2 )
(b)
f ( x + 2) − 2 f ( x + 1) + f ( x) = 0
(d)
sin nx
is 4π,is
sin x / n
4
28
Punjab EDUSAT Society (PES)
MATHEMATICS
(c)
f ( x ) + f ( x + 1) = f ( x 2 + x )
(d)
x + x2
f ( x1 ) + f ( x2 ) = f 1
1 + x1 x2
9.
x−m
, where m ≠ n ⋅ Then
x−n
f is one-one into
f is many one into
Let f : R → R be a function defined by f ( x ) =
(a)
(c)
10.
(b)
f is one-one onto
f is many one onto (d)
1
f ( x ) = 2 the function is
x
(a)
symmetric about x-axis
(b)
symmetric about
y-axis
11.
12.
(c)
symmetric about x and y-axis
(d)
none of these.
If f ( − x ) = − f ( x ), then f ( x ) is
(a)
an even function
(b)
an odd function
(c)
neither odd nor even
(d)
periodic function
The function f ( x ) = x − [ x ] , ([] represent greatest integer function), is
(a)
a constant function
(b)
periodic with period ½
(c)
periodic with period 1
(d)
none periodic function
x2 + 1
If g : [ −2, 2] → R where g(x) = x3 + tan x +
(where [x] denotes
p
greatest
integer ≤ x) be an odd function, then the value of the parameter p is
(a)
-5 < p < 5
(b)
p<5
(c)
p>5
(d) none of
these.
−1 if x < 0
14.
Let g(x) = 1 + x – [x] and f ( x ) = 0 if x = 0 then ∀x, fog(x) equals
1 if x > 0
(a)
x
(b)
1
(c)
f(x)
(d)
g(x)
−1
cos x
15.
The domain of the function f ( x ) =
is:
[ x]
13.
(a)
[-1,0) ∪ {1}
(b)
[-1, 1]
(c)
[-1, 1) (d)
None of
these
16.
If φ(x) is the inverse of the function f(x) and f ′ ( x ) =
(a)
1
1 + {φ ( x )}
5
(b)
1
1 + { f ( x )}
5
(c)
1
d
, then
φ ( x ) is
5
dx
1+ x
1 + {φ ( x )}
5
(d)
1+
f(x)
17.
x
The domain of sin −1 log 3 is
3
(a)
29
Punjab EDUSAT Society (PES)
MATHEMATICS
[1,9]
18.
19.
(b)
[ - 1, 9 ]
(c)
[ − 9,1]
(d)
[ -9, -1 ]
Let f : N → N defined by f ( x) = x + x + 1, x ∈ N , then f is
(a)
One one onto
(b)
Many one onto
(c)
One-one but not onto
(d)
None of the above
2
x
The domain of sin −1 log 3 is
3
(a)
[1, 9]
(b)
[-1, 9]
(c)
[-9, 1]
(d)
(b)
(d)
an even function
a periodic function
[-9,
-1]
20.
(
(a)
(c)
21.
neither an even nor odd function
an odd function
3
+ log10 ( x3 − x ) , is
2
4− x
(-1, 0) ∪ (1, 2) ∪ (2, ∞)
(b)
(a, 2)
(-1, 0) ∪ (a, 2)
(d)
(1, 2) ∪ (2, ∞)
Domain of the function f ( x ) =
(a)
(c)
22.
)
The function f ( x ) = log x + x 2 + 1 , is
If f: R → R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and f(1) = 7, then
n
∑ f ( r ) , is
r =1
(a)
7 n ( n + 1)
2
(b)
7n
2
(c)
7 ( n + 1)
2
(d)
7n + (n +
1)
23.
Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2,
3, 4}.
The relation R is
(a)
reflexive
(b)
transitive
(c)
not symmetric (d)
a
function
24.
25.
The range of the function f ( x ) = 7− x Px −3 is
(a)
{1, 2, 3}
(b)
{3, 4, 5, 6}
(c)
{1,2,3,4,5}
{1, 2, 3, 4}
Let f : (-1, 1) → B, be a function defined by f ( x ) = tan −1
one-one
(a)
and onto when B is the interval
π
π
(c)
0, (b)
0, 2
2
π π
− 2 , 2
(d)
2x
, then f is both
1 − x2
(d)
π π
− ,
2 2
26.
A function is matched below against an interval where it is supposed to be
increasing
Which of the following pairs is incorrectly matched
30
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
Interval
( −∞, ∞ )
Function
x3 – 3x2 + 3x +3
(b)
[ 2,∞ )
2x3 – 3x2 – 12x + 6
(c)
27.
28.
1
−∞,
3
( −∞, −4]
3x2 – 2x + 1
(d)
x3 + 6x2 + 6
A real valued function f(x) satisfies the functional equation
f ( x − y ) = f ( x ) f ( y ) − f ( a − x ) f ( a + y ) where a is a given constant and
f ( 0 ) = 1, f ( 2a − x ) is equal to
(a)
-f(x) (b)
f(x)
(c)
f(a) + f(a – x) (d)
f(-x)
−π π
The largest interval lying in
, for which the function .
2 2
− x2
−1 x
f ( x) = 4 + cos 2 − 1 + log (cos x) is defined, is
π π
π
π π
(a)
(b)
[− , )
(c)
[ 0, )
− ,
4 2
2
2 2
(d) [0, π )
Let f : N → Y be a function defined as f ( x) = 4 x + 3
where Y = { y ∈ N : y = 4 x + 3 for some x ∈ N }.
Show that f is invertible and its inverse is
y+3
y −3
3y + 4
(b)
(c) g ( y ) =
(d)
(a) g ( y ) =
g ( y) =
4
4
3
y+3
g ( y) = 4 +
4
30.
Let R be the set of real numbers. If F : R → R is a function defined by
f ( x ) = x2
then f is
(a)
injective but not surjective (b)
surjective but not injective
(c)
bijective
(d)
none of these.
π
π
31. If the values of the function f(x) = cos x – x(1 + x) ≤ x ≤ lies between a
3
6
and b then
1 π π
3 π π
(a)
a = − 1 + , b =
− 1 +
2 3
3
2 6
6
29.
(b)
(c)
(d)
1 π π
3 π π
− 1 + , b =
− 1 +
2 6
6
2 3
3
3 π π
1 π π
a=
− 1 + , b = − 1 +
2 3
3
2 6
6
none of these
a=
31
Punjab EDUSAT Society (PES)
MATHEMATICS
32.
33.
34.
35.
( x + 1)( x − 3)
f ( x ) =
then the domain of f(x) must be
x
−
2
(
)
(a)
(b)
] − ∞, −1] ∪ [3, ∞[
] − ∞, −1] ∪ [2,3]
(d)
none of these
(c)
[ −1, 2[∪[3, ∞[
Let f be an injective map with domain {x, y, z} and range {1, 2, 3} such that
exactly one the three statements is correct and the remaining two are false
f(x) = 1, f(y) ≠ 1, f(z) ≠ 2, then value of f -1(1) is
(a)
x
(b)
y
(c)
z
(d)
none
If p, q, r are any real numbers, then
1
(a)
max (p, q) < max (p, q, r)
(b)
min (p, q) = ( q − p − p − q )
2
(c)
max (p, q) < min (p, q, r)
(d)
none of these.
If f(x) = cos(log x), then f(x) f(y) =
(a)
½ [f(y/x) + f(xy)
(b)
½ [f(x/y) + f(xy)]
(c)
36.
37.
½ [f(xy) + f(y)]
(d)
none
If the function f(x) and g(x) are defined on R → R such that
0, x ∈ rational
0, x ∈ irrational
f ( x) =
,
g ( x) =
, then (f – g) (x) is
x, x ∈ irrational
x, x ∈ rational
(a)
one-one and onto
(b)
neither one-one nor onto
(c)
one-one but not onto
(d)
onto but not one-one
X and Y are two sets and f : X → Y, if {f(c) = y; c ⊂ X, y ⊂ Y},
{f -1(d) = x, d ⊂ Y, x ⊂ X}, then the true statement is
f −1 ( f ( a ) ) = a
(a)
(b)
f ( f −1 ( b ) ) = b
(c)
f ( f −1 ( b ) ) = b, b ⊂ y
(d)
f −1 ( f ( a ) ) = a, a ⊂ x
Answers( Function)
1.
b
2.
c
3.
a, b
4.
b
5.
a
6.
c
7.
c
8.
d
9.
b
10.
b
11.
b
12.
c
13.
c
14.
b
15.
a
16.
c
17.
a
18.
a
19.
a
20.
c
21.
a
22.
a
23.
c
24.
a
25.
c
26.
c
27.
a
28.
c
29.
b
30.
d
31.
a
32.
c
33.
b
34.
b
35.
b
36.
a
32
Punjab EDUSAT Society (PES)
MATHEMATICS
37.
d
Limit, Continuity and Differentiability
1.
x , if x is rational
If f ( x ) =
, then
1 − x, if x is irrational
(a)
f is only right continuous at x = ½
(b)
f is only left continuous at
x=½
(c)
at all points.
f is continuous at x = ½
(d)
f is discontinuous
1 − cos8 x
=
x → 0 1 − cos 6 x
(a)
64/36 (b)
5/8
(c)
64/25
(d)
15/23
3.
If f(x) = x sin 1/x, x = 0, then value of function at x = 0, so that the function is
continuous at x = 0, is
(a)
1
(b)
0
(c)
-1
(d)
indeterminate
cot x
4.
In order that the function f(x) = (x + 1)
is continuous at x = 0, f(x) must be
defined as
(a)
f(0) = 0 (b) f(0) = e
(c)
f(0) = 1/e
(d)
none of
these
2.
lim
5.
If f(x) and f ( x ) are both differentiable, then f(x) is
(a)
(c)
only left continuous at x = 0 (b)
not continuous at x = 0
only right continuous at x = 0
(d)
continuous every where
33
Punjab EDUSAT Society (PES)
MATHEMATICS
6.
7.
8.
9.
10.
x
2 − 1, 0 ≤ x ≤ 1
Let f ( x ) =
, g(x) = (2x + 1)(x – k) + 3, 0 ≤ x ≤ ∞, then
1 , 1≤ x ≤ 2
2
g[f(x)] is continuous at x = 1, if k equals
(a)
½
(b)
11/6 (c)
1/6
(d)
13/6
n
1
n
Evaluate the lim 2
+ 2
+ ....... +
2
n →∞ n + 1
n +2
2n
(a)
π
(b)
π/3
(c)
π/2
(d)
π/4
log x − 1
lim
=
x →e
x−e
(a)
1
(b)
½
(c)
1/e
(d)
0
1+ x − 3 1+ x
The value of f(0), so that the function f ( x ) =
is continuous at
x
x = 0, is
(a)
2
(b)
¼
(c)
1/6
(d)
1/3
The set of all points where the function f(x) = x x, is differentiable is
(a)
(c)
( −∞,0 ) ∪ ( 0, ∞ ) (b) ( −∞,0 )
( −∞, ∞ )
( 0,∞ )
11.
12.
(
(d)
)
sin π − x
3
is equal to
limπ
x → 3 ( 2cos x − 1)
2
1
(b)
(a)
3
3
(c)
3
(d)
½
2 x − 3 ⋅ [ x], x ≥ 1
The function f ( x ) = πx
sin 2 , x < 1
(where [x ] denotes greatest integer ≤ x)
(a)
is continuous at x = 2
(b) is differentiable at x = 1
(c)
is continuous but not differentiable at x = 1 (d) none of these
( 27 − 2 x ) 3 − 3 x ≠ 0
f ( x) =
)
1 (
9 − 3 ( 243 + 5 x ) 5
1
13.
14.
=4
The value of f(0), so that the function
is
continuous is given by
(a)
2/3
(b)
6
(c)
2
(d)
4
If f(x), g(x) be twice differentiable functions on [0, 2], satisfying
f ′′ ( x ) − g ′′ ( x ) , f ′ (1) = 2, g (1) = 4 and f(2) = 3, g(2) = 9, then f(x) – g(x) at x
equals
(a)
0
(b)
-10
(c)
8
(d)
2
34
Punjab EDUSAT Society (PES)
MATHEMATICS
15.
16.
1
1
1
1
lim +
+
+ ....... + is equal to
nb
na na + 1 na + 2
(a)
log(b/a)
(b)
log(a/b)
(c)
n →∞
The value of lim
18.
19.
0
(a)
e
(c)
1
n
(a)
x = 2 only
(d)
log b
is equal to
2
(d)
none of these
is equal to
x
(c)
y
(d)
none of these
x + 2 , 1 ≤ x ≤ 2
The function f ( x ) = 4
,
x = 2 is continuous at
3x − 2,
x>2
(b)
x 3 2 − 27
=
lim
x →9
x − 9
(a)
3/2
lim
x →0
these
22.
-1
n →∞
(a)
21.
(b)
If 0 < x < y, then lim ( y n + x n )
(1 + x )
20.
x2
x →∞
(a)
17.
1 + x 4 − (1 + x 2 )
log a
limπ
1
x
−e+
x2
11e
24
[
1 − tan x
(b)
x ≤ 2 (c)
x > 2 (d)
(b)
9/2
(c)
none of these
2/3
(d)
1/3
ex
2 =
(b)
−11e
24
(c)
e
24
(d)
none of
1 − 2 sin x
(a)
sin 1 (b)
2
(c)
π
(d)
3
If f(x) = sin x, then f(x) is not differentiable at
(a)
x = 0 only
(b)
all x
(c)
multiples of π
(d)
multiples of π/2
x→ 4
n
a
23.
lim 1 + sin equals to
n →∞
n
a
(a)
e
(b)
e
(c)
e2a
(d)
0
24.
Function f(x) = (x - 1+x - 2+ cos x) where x ∈ [0, 4] is not continuous at
number
of points
(a)
1
(b)
2
(c)
3
(d)
0
1
1
1
1
lim
25.
+
+
+ ....... +
n →∞ 1.3
3.5 5.7
( 2n − 1) .( 2n + 1)
(a)
½
(b)
1/3
(c)
¼
(d)
none of these
35
Punjab EDUSAT Society (PES)
MATHEMATICS
26.
27.
eax − ebx
equals
x →0
x
(a)
a + b (b)
a – b (c)
eab
(d)
1
1
p
x sin , x ≠ 0
Let f ( x ) =
then f(x) is continuous but not differential at x = 0
x
0
, x=0
The value of lim
if
(a)
0<p≤1
(b)
1≤p<∞
(c)
-∞ < p < 0
(d)
p=
0
1− x
, x ≠−1
If f ( x) = 1+ x
then f([2x]) (where [] represent greatest integer
28.
1 , x =−1
function),is
(a)
continuous at x = -1
(b)
continuous at x = 0
(c)
discontinuous x = ½
(d)
all of these
tan
x
−
cot
x
π
, x≠
π
4
π
x−
29.
Let f ( x) =
the value of a so that f(x) is continuous at x =
4
4
π
, x=
a
4
is:
(a)
2
(b)
4
(c)
3
(d)
1
−1
−1
−1
cos ec ( sec α ) + cot ( tan α ) + cot cos ( sin −1 α )
30.**The value of lim
α→ 0
α
(a)
is 0
(b)
is -1 (c)
is -2 (d)
does not exist
−1, x < 0
31.
Let g(x) = 1 + x – [x] and f ( x ) = 0 , x = 0 then for all x, f[g(x)] is equal to
1 , x > 0
(a)
x
(b)
1
(c)
f(x) (d)
g(x)
32.
(a + h)
lim
h
2a sin a
a2 cos a + 2a sin a
If f(x) = x
(a)
(b)
(c)
34.
sin ( a + h ) − a 2 sin a
h →0
(a)
(c)
33.
2
(
)
(b)
(d)
is equal to
a2 cos a
none of these.
x + 1 + x , then
f(x) is continuous but not differentiable at x = 0
f(x) is differentiable at x = 0
f(x) is not differentiable at x = 0
(d) none of these
The value of lim
x →0
1 + sin x − cos x + log (1 − x )
x3
is
36
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
35.
-1
The value of lim
x→0
4
( ln 4 )2
3
(a)
36.
½
(4
x
)
−1
(c)
-1/2
(d)
(c)
3
( ln 4 )2
2
2
x
sin log (1 + 3x )
4
4
3
(b) ( ln 4 )
3
is
3
( ln 4 )3
2
2 f ( x) − 3 f ( 2x) + f ( 4x)
(d)
x2
x →0
to
3a
(b)
1
3
If f(x) is differentiable function and f′′(0) = a, then lim
is equal
(a)
37.
(b)
2a
(c)
5a
(d)
4a
if x is rational
x,
then the number of points at which
1 − x, if x is irrational
If function f ( x ) =
f(x) is continuous, is
(a)
∞
(b)
1
(c)
0
(d)
none of these
n
n
38. Let f(a) = g(a) = k and there nth derivatives f (a), g (a) exist and are not equal
f (a) g ( x) − f (a) − g (a) f ( x) + g (a)
for some n. Further if lim
= 4 then the
x →a
g ( x) − f ( x)
value of k is
(a)
0
(b)
1
(c)
2
(d)
4
1
x + 5x + 3 x
lim 2
x →∞ x + x + 3
(a)
0
(b)
4
(c)
2
(d)
1
x
1 − tan 2 [1 − sin x ]
is
limπ
x→ 2
3
x
1 + tan 2 [ π − 2 x ]
(a)
∞
(b)
1/8
(c)
0
(d)
1/32
1 − tan x
π
π
Let f ( x ) =
, x ≠ , x ∈ 0, . If f(x) is continuous in
4x − π
4
2
2
39.
40.
41.
π
f is
4
(a)
-1
(b)
½
(c)
-½
(d)
π
0, 4 , then
1
2x
42.
43.
44.
a b
If lim 1 + + 2 = e 2 , then the values of a and b, are
x →∞
x x
(a)
a = 1 and b = 2
(b)
a = 1, b ∈ R
(c)
a ∈ R, b = 2
(d)
a ∈ R, b ∈ R
1
Suppose f(x) is differentiable at x = 1 and lim (1 + h ) = 5 , then f ′(1) equals
h→0 h
(a)
3
(b)
4
(c)
5
(d)
6
Let f be differentiable for all x. If f(1) = -2 and f ′(x) ≥ 2 for x ∈ [1, 6], then
(a)
f(6) ≥ 8
(b)
f(6) < 8
37
Punjab EDUSAT Society (PES)
MATHEMATICS
(c)
45.
46.
x→ α
(a)
48.
(x
− α
a2
2
( α − β)
2
)
2
(b)
is equal to
−a 2
2
(c)
( α − β ) (d)
2
0
(-∞, ∞)
1
2
( α − β)
2
(d)
½ tan 1
x
is differentiable is
1+ x
(b) (0, ∞) (c)
(-∞, 0) ∪(0, ∞)
(d) (-∞, -1)∪(-1,
Let f : → be a function defined by f ( x) = Min {x + 1, x + 1}. Then which
of the following is true ?
(a) f ( x) is not differentiable at x = 1 (b) f ( x) is differentiable every
f ( x) ≥1 for all x ∈
1
2
The function f : \ {0} → given by f ( x) = − 2 x
can be made
x e −1
continuous at x = 0 by defining f (0) as
(a)
-1
(b)
0
(c)
1
(d)
2
(c)
50.
1 − c o s (a x 2 + b x + c )
The set of points where f ( x ) =
∞)
where .
f(6) = 5
1
2
4
1
1
lim 2 sec 2 2 + 2 sec2 2 + ...... + sec2 1 equals
x →∞ n
n
n
n
n
(a)
½ sec 1
(b)
½ cosec 1
(c)
tan 1
(a)
49.
(d)
If f is a differentiable function satisfying f(x) – f(y)≤ ( x – y)2, x, y ∈ R and
f(0) = 0, then f(1) equals
(a)
-1
(b)
0
(c)
2
(d)
1
2
Let α and β be the distinct roots of ax + bx + c = 0, then
li m
47.
f(6) < 5
f ( x) is not differentiable at x = 0 (d)
51.
The first two terms of a geometric progression add up to 12. The sum of the
third and the fourth terms is 48. If the terms of the geometric progression are
alternately positive and
negative, then the first term is
(a)
12
(b)
4
(c)
-4
(d)
-12
1
if x ≠ 1
( x − 1) sin
52.
Let f ( x) =
. Then which one of the following is
x −1
0
if x = 1
true?
(a)
f is differentiable at x = 0 but not at x = 1
(b)
f is differentiable at x = 1 but not at x = 0
(c)
f is neither differentiable at x = 0 nor at x = 1
(d)
f is differentiable at x = 0 and n at x = 1
38
Punjab EDUSAT Society (PES)
MATHEMATICS
53.
(a)
54.
x − sin x
, then lim f ( x ) is
x →∞
x + cos 2 x
(b)
∞
(c)
1
If f ( x ) =
0
(d)
none of these
For a real number y, let [y] denote the greatest integer less than or equal to y.
Then the function f ( x ) =
tan ( π [ x − π])
(a)
(b)
is
2
1 + [ x]
discontinuous at some x
continuous at all x, but the derivative f ′(x) does not exist for some x
(c)
f ′(x) exists for all x, but the derivative f ″(x) does not exist for some x
f ″(x) exist for all x
1
2
− x , if x ≠ 1
( x − 1) sin
Let f ( x ) =
be a real valued function. Then
( x − 1)
−1
, if x = 1
set of points where f(x) is not differentiable is
(a)
{0}
(b)
{0, 1} (c)
{, 1, -1}
(d) none of these
( x3 + x2 +16x + 20)
− x , if x ≠ 2
2
Let f ( x) =
. If f(x) is continuous for all x, then
( x − 2)
,
if x = 2
k
(d)
55.
the
56.
k=
57.
58.
59.
p be
(a)
7
(b)
2
(c)
If lim f ( x ) g ( x ) exists then
x →a
0
(d)
-1
(a)
both lim f ( x ) and lim g ( x ) exist
(b)
both lim f ( x ) and lim g ( x ) do not exist
(c)
both lim f ( x ) and lim g ( x ) may not exist (d)
lim
x →0
2x − 1
(1 + x )
1
2
x →a
x →a
x →a
x →a
x →a
x →a
none of these
=
−1
(a)
log 4 (b)
log 2
(c)
1
(d)
0
n
( x − 1)
Let g ( x) =
;0 < x < 2, m and n are integers , m ≠ 0, n > 0, and let
log cos m ( x − 1)
the left
hand derivative of | x – 1| at x = 1. If lim+ g ( x) = p, then
x →1
(a)
n = 1, m = 1
n > 2, m = n
(b)
n = 1, m = −1 (c) n = 2, m = 2 (d)
39
Punjab EDUSAT Society (PES)
MATHEMATICS
60.
Let f(x) be a non-constant twice differentiable function defined on
1
(−∞, ∞) such that f ( x) = f (1 − x) and f ′ = 0. Then ,
4
1
(a)
f ′′( x) vanishes at least twice on [ 0 , 1 ]
(b)
f ′ = 0
2
1/ 2
1
(c)
f x + sin xdx = 0
(d)
∫
2
−1/ 2
1/ 2
∫
1
f (t )e
sin πt
dt =
0
61.
∫
f (1 − t ) esin πi dt
1/ 2
1
If f(x) is continuous and differentiable function and f = 0∀n ≥ 1 and n ∈ I,
n
then
(a)
(c)
f(x) = 0, x ∈ (0, 1]
(b)
f ′(0) = 0 = f ′′(0), x ∈ (0, 1] (d)
f(0) = 0, f ′(0) = 0
f(0) = 0 and f ′(0) need not be
zero
62.
( )
sin x
1
For x > 0, lim ( sin x ) x + 1
is
x
x →0
(a)
0
(b)
-1
(c)
1
(d)
2
ANSWERS(Limit, Continuity and Differentiability)
1.
7.
13.
19.
c
d
c
b
2.
8.
14.
20.
a
c
b
a
3.
9.
15.
21.
b
c
a
b
4.
10.
16.
22.
b
c
a
a
5.
11.
17.
23.
d
b
c
a
6.
12.
18.
24.
a
c
c
d
40
Punjab EDUSAT Society (PES)
MATHEMATICS
25.
31.
37.
43.
49.
55.
61.
a
b
c
c
d
d
a,b,c,d
b
26.
32.
38.
44.
50.
56.
b
c
d
a
c
a
62.
c
27.
33.
39.
45.
51.
57.
a
a
b
b
c
c
28.
34.
40.
46.
52.
58.
d
c
d
a
a
a
29.
35.
41.
47.
53.
59.
b
b
c
d
c
c
30.
36
42.
48.
54.
60.
d
a
b
a
d
DIFFERENTIATION
1.
If y = sin x + sin x + sin x + ........∞ , then the value of dy/dx =
(a)
2.
3.
sin x
y +1
(b)
sin x
(c)
y +1
The derivative of f(x) = x at x = 0 is
(a)
1
(b)
0
(c)
-1
If xy . yx = 1, then dy/dx =
cos x
(d)
2y +1
(d)
cos x
2y −1
it does not exist
41
Punjab EDUSAT Society (PES)
MATHEMATICS
y ( x log y − y )
(a)
If
(x
2
(c)
8.
9.
10.
( y / x)
ae
y ( x log y − y )
x ( y log x + x )
−
y ( x log y + y )
x ( y log x + x )
, a > 0 . Then y″(0), equals
π
2
(c)
2 −π
− ⋅e 2
a
(d)
a −π2
⋅e
2
2
h 2 + ab
( hx + by )
(d)
3
h 2 − ab
( hx + by )
2
h 2 − ab
( hx + by )
3
a x ............. ∞
dy
the function y = a x
is
dx
y2
y 2 log y
(b)
x (1 − y log x )
x (1 − y log x )
The equation of
(a)
7.
−1
d2y
If ax + 2hxy + by = 1, then
equals
dx 2
h 2 + ab
(b)
(a)
2
( hx + by )
2
(c)
6.
+ y 2 ) = ae tan
a π2
⋅ e (b)
2
(a)
5.
x ( y log x − x )
(d)
y ( x log y + y )
(c)
4.
x ( y log x − y )
(b)
y 2 log y
x (1 − y log x log y )
(d)
y 2 log y
x (1 + y log x log y )
d2 y
is
dx 2
(a)
a constant
(b)
a function of x only
(c)
a function of y only (d)
a function of x and y
dy
If x = a(cos θ + θ sin θ), y = a(sin θ - θ cos θ, then
=
dx
(a)
cos θ (b)
tan θ (c)
sec θ (d)
cosec θ
If f(x) = logx (log x), then f′(x) at x = e is
(a)
0
(b)
1
(c)
1/e
(d)
½e
The differentiable coefficient of f(log x) w.r.t x, where f(x) = log x is
(a)
x/log x (b)
(log x)/x
(c)
(x log x)-1
(d)
none of
If y2 = ax2 + bx + c, then y 3 ⋅
these
11.
12.
13.
x∞
dy
is
dx
(a)
x2
(b)
y2
(c)
xy2
(d)
If f(x) = x + 2, then f′(f(x)) at x = 4 is
(a)
8
(b)
1
(c)
4
(d)
1 + sin x
π
π
If f ( x ) = tan −1
, 0 ≤ x ≤ , then f ′
1 − sin x
2
6
(a)
-1/4 (b)
-1/2 (c)
¼
(d)
If y = x x
xx
, then x(1 – y log x)
none of these
5
is
½
42
Punjab EDUSAT Society (PES)
MATHEMATICS
14.
(a)
15.
16.
17.
dy
is equal to
dx
2x + 2 y
(b)
(c)
1 + 2x+2
If 2x + 2y = 2x + y, then
2x + 2 y
2x − 2 y
(2
x− y
2y −1
) ⋅ 1 − 2 x (d)
2 x+ y − 2 x
2y
dy
1 − ln x
If y = cos −1
at x = e is
, then
dx
1 + ln x
(a)
- 1/e (b)
-1/2e (c)
1/2e (d)
1/e
If y is a function of x and log(x + y) = 2xy, then the value of y′(0) is equal to:
(a)
1
(b)
-1
(c)
2
(d)
0
f ′ (1) f ′′ (1) f ′′′ (1)
( −1)′′ f ′′ (1)
If f(x) = xn then the value of f (1) −
+
+
+ .......
1!
2!
3!
n!
is
(a)
18.
1
(b)
2n
(c)
2n – 1
The solution of the differential equation (1 + y2) + x − e tan
−1
(a)
xe2 tan
(c)
2 xe tan
−1
y + ..... to∞
−1
y
= e tan
y
= e2 tan
y
−1
+k
y
+k
( x − 2 ) = ke tan
(d)
xe tan
20.
1+ x
1− x
(b)
1/x
(c)
x
x
m+ n
m
n
If x ⋅ y = ( x + y ) , then dy/dx is
(a)
(a)
22.
23.
24.
xy
(b)
x/y
−1
y
−1
x
) dydx = 0
y
= tan −1 y + k
dy
is
dx
If x = e y + e
, x > 0, then
0
−1
(b)
19.
21.
(
(d)
(c)
y/x
(d)
x
1+ x
(d)
x+ y
xy
If xexy = y + sin2 x, then at x = 0, dy/dx =
(a)
1
(b)
2
(c)
4
(d)
3
If x2 + y2 = 1, then
(a)
yy″ - 2(y′)2 + 1 = 0
(b)
yy″ + (y′)2 + 1 = 0
2
(c)
yy″ - (y′) – 1 = 0
(d)
yy″ + 2(y′)2 + 1 = 0
If y is function of x and log(x + y) – 2xy = 0, then the value of y′(0) is equal to
(a)
1
(b)
-1
(c)
2
(d)
0
If f ′′ ( x ) = − f ( x ) , g ( x ) = f ′ ( x ) , F ( x ) =
2
2
x x
f + g and F(5) = 5,
2 2
then
25.
F(10) is equal to
(a)
5
(b)
2
d x
equals
dy 2
10
(c)
0
(d)
15
43
Punjab EDUSAT Society (PES)
MATHEMATICS
1.
7.
13.
20.
−1
−1
(a)
d2 y
2
dx
(c)
d 2 y dy
2
dx dx
d
a
d
c
2.
8.
14.
21.
d
b
c
a
−3
(b)
d 2 y dy
− 2
dx dx
d 2 y dy
− 2
dx dx
−3
(d)
5.
11.
18.
24.
6.
12.
19.
25.
−2
ANSWERS(Differentiation)
3.
d
4.
c
9.
b
10.
c
16.
a
17.
d
22.
b
23.
a
d
c
c
a
c
b
c
b
44
Punjab EDUSAT Society (PES)
MATHEMATICS
Application of Derivative – I
1.
The value of k in order that f(x) = sin x – cos x – kx + b decreases for all real
values, is given by
(a)
k < 1 (b)
k > 1 (c)
(d)
2.
3.
f(x) = x(x – 2) (x – 4), will satisfy mean value theorem at
(a)
1
(b)
2
(c)
3
(d)
4
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if
45
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
k ≥ 3/2
(b)
k > 3/2 (c)
k < 3/2 (d)
k ≤ 3/2
4.
The set of all values of a for which the function
a+4 5
for all real x, is
f ( x ) =
− 1 x − 3 x + log 5 decreases
1− a
(a)
(-∞, ∞)
(b)
(1, ∞)
5 − 27
3 − 21
(c)
(d)
−3,
∪ ( 2, ∞ )
−4,
∪ (1, ∞ )
2
2
π
π
5.
For the given integer k, in the interval 2π k − , 2π k + the graph of sin x
2
2
is
(a)
increasing from -1 to 1
(b)
decreasing from -1 to 0
(c)
decreasing from 0 to 1
(d)
none of these
6.
The length of the sub tangent to the curve x2 + xy + y2 = 7 at (1, -3) is
(a)
3
(b)
5
(c)
15
(d)
3/5
7.
If f(x) = 2x6 + 3x4 + 4x2 then f′(x) is
(a)
even function
(b)
(c)
neither even nor odd
(d)
an odd function
none of these
8.
Normal at a point to the parabola y2 = 4ax, when abscissa is equal to ordinate,
will meet the parabola again at a point
(a)
(6a, -9a)
(b)
(-9a, 6a)
(c)
(-6a, 9a)
(d)
(9a,
-6a)
9.
The angle between the tangents drawn from the point (1, 4) to the parabola y2
= 4x is:
(a)
π/6
(b)
π/4
(c)
π/3
(d)
π/2
10.
For what value of a, f(x) = -x2 + 4ax2 + 2x – 5 is decreasing ∀x.
(a)
(1, 2)
(b)
(3, 4)
(c)
R
value of a
11.
The common tangent of the parabolas y2 = 4x and x2 = -8y is
(d)
no
(a)
y=x+2
(b)
y=x–2
(c)
y = 2x + 3
(d)
none of these
12.
The function f ( x) = x( x + 3)e− (1/ 2) x satisfies all the conditions of Roll’s
theorem in [−3, 0].
The value of c is
(a)
0
(b)
-1
(c)
-2
(d)
-3
46
Punjab EDUSAT Society (PES)
MATHEMATICS
13.
e2 x − 1
f ( x) = 2 x is
e +1
(a)
an increasing (b)
a decreasing
(c) an even
(d)
None of these
14.
A point on the parabola y2 = 18x at which ordinate increases at twice the rate
of abscissa is
9 9
−9 9
(a)
(b)
(c)
(d)
(2,
( 2, − 4 )
,
,
8 2
8 2
4)
15.
A function y = f(x) has a second order derive f″(x) = 6(x – 1). If its graph
passes through the point
(2, 1) and at that point the tangent to the graph is y = 3x
– 5, then the function is
(b)
(x – 1)2
(c)
(x + 1)3
(d)
(x –
(a)
(x + 1)2
1)3
16.
The normal of the curve x = a(1 + cos θ), y = a sin θ at ‘θ’ always passes
through the fixed point
(a)
(a, a) (b)
(0, a) (c)
(0, 0) (d)
(a, 0)
17.
The normal to the curve x = a (cos θ + θ sin θ), y = a(sin θ - θ cos θ) at any
point ‘θ’ is such that
(a)
it passes through the origin
(b)
it makes angle π/2 + θ
with the x-axis
π
(c)
it passes through a , θ − α
(d)
it is constant
2
distance from the origin
18.
a spherical iron ball 10 cm in radius is coated with a layer of ice of uniform
thickness that melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then
the rate at which the thickness
of ice decreases, is
1
1
(a)
(b)
cm / min .
cm / min
36π
18π
1
5
(c)
(d)
cm / min
cm / min
54π
6π
19.
If the equation an x n + an −1 x n −1 + ...... + a1 x = 0, a1 ≠ 0, n ≥ 2 , has a positive root
x = α, then the
equation nan x n−1 + ( n − 1) an −1 x n −2 + .....a1 = 0 has a positive root,
which is
(a)
greater than α
(b)
smaller than α
(c)
greater than or equal to α
(d)
equal to α
47
Punjab EDUSAT Society (PES)
MATHEMATICS
20.
0) is
Angle between the tangents to the curve y = x2 – 5x + 6 at points (2, 0) and (3,
(a)
π/6
(b)
π/4
(c)
π/3
(d)
π/2
21.
The function f ( x) = tan −1 (sin x + cos x) is an increasing function in
22.
(a)
(−π / 2, π / 4) (b) (0, π / 2) (c)
Consider the following statements S and R
(−π / 2, π / 2) (d)
(π / 4, π / 2)
π
S: Both sin x and cos x are decreasing functions in the interval , π
2
R: If a differentiable function decreases in the interval (a, b), then its
derivative also decreases in (a, b). Which of the following is true?
(a)
Both S and R are wrong
(b)
Both S and R are correct, but R is not the correct explanation of S
23.
(c)
S is correct and R is the correct explanation for S
(d)
S is correct and R is wrong
If f(x) = xex(1 – x), then f(x) is
(a)
increasing on [- ½ , 1]
(b)
decreasing on IR
(c)
increasing on IR
(d)
decreasing on [- ½ , 1]
24.
The triangle formed by the tangent to the curve f(x) = x2 + bx – b at the point
(1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the
value of b is
(a)
-1
(b)
3
(c)
-3
(d)
1
25.
The point(s) on the curve y3 + 3x2 = 12y where the tangent is vertical, is (are)
11
4
4
, −2 (b)
, 2
(a)
(c)
(0, 0) (d)* ±
,1
±
±
3
3
3
26.
The length of a longest interval in which the function 3sin x – 4sin3 x is
increasing, is
(a)
π/3
(b)
π/2
(c)
3π/2 (d)
π
27.
In [0, 1] Lagranges Mean Value theorem is NOT applicable to
1
1
x<
sin x
2 − x,
2
, x≠0
(a)
f ( x) =
(b)
f ( x) = x
2
1 − x , x ≥ 1
1 , x = 0
2
2
28.
(c)
f(x) = xx
(d)
f(x) = x
3
2
2
If f(x) = x + bx + cx + d and 0 < b < c, then in (-∞, ∞)
(a)
f(x) is a strictly increasing function (b)
f(x) has a local maxima
(c)
f(x) is a strictly decreasing function (d)
f(x) is bounded
48
Punjab EDUSAT Society (PES)
MATHEMATICS
29.
If f(x)= xα log x and f(0) = 0, then the value of α for which Rolle’s theorem
can be applied in
[0, 1] is
(a)
-2
(b)
-1
(c)
0
(d)
½
x
The tangent to the curve y = e drawn at the point (c, ec) intersects the line
30.
joining the
points (c – 1, ec-1) and (c + 1, ec+1)
(a)
(c)
(b)
(d)
on the left of x = c
at no point
on the right of x = c
at all points
ANSWERS(Application of Derivative-I)
14.
c
15.
c
16.
b
17.
d
18.
a
19.
c
20.
13
b
a
21.
14.
d
a
22.
15.
d
b
23.
16.
d
d
24.
17.
d
d
25.
18.
c
b
19.
25.
b
d
20.
26.
d
a
21.
27.
a
a
22.
28.
d
a
23.
29.
a
d
24.
30.
c
a
49
Punjab EDUSAT Society (PES)
MATHEMATICS
Application of Derivatives – II (Maxima and Minima)
The least perimeter of an isosceles triangle in which a circle of radius r can be
1.
inscribed is
(a)
4 3r (b)
2 3r
(c)
8 3r
6 3 r (d)
The minimum value of x − 3 + x − 2 + x − 5 is
2.
(a)
3
(b)
7
(c)
15
(d)
0
2
x − x +1
3.
For real x, maximum value of 2
is
x + x +1
(a)
½
(b)
2
(c)
1/3
(d)
3
3
2 2
4.
If the function f(x) = x + α x + βx + 1 has maximum value at x = 0 and
minimum at x = 1, then
(a)
α = 2/3, β = 0 (b)
α = -3/2, β = 0 (c)
α = 0, β = 3/2 (d) none
of these
2
x 2 t − 5t + 4
dt . The point of extrema of the function in the
5.
f(x) = f ( x ) = ∫
0
2 + et
interval (1, 3) is
(a)
x = 1 (b)
x = 2 (c)
x = 2.1
(d)
x = 1.5
6.
The greatest value of f(x) = 2 sin x + sin 2x, on [0, 3π/2], is given by
3 3
(a)
9/2
(b)
5/2
(c)
(d)
3/2
2
7.
Let f(x) = cos x sin 2x. Then
(a)
min {f(x); (-π ≤ x ≤ π) > - 8/9}
(b)
min {f(x); (-π ≤ x ≤ π) > 3/7}
(c)
min {f(x); (-π ≤ x ≤ π) > - 1/9}
(d)
min {f(x); (-π ≤ x ≤ π) > 2/9}
8.
The maximum value of 12 sin θ - 9 sin2 θ is
(a)
3
(b)
4
(c)
5
(d)
none of these.
9.
What are the minimum and maximum values of the function f(x) = x5 – 5x4 +
5x3 – 10?
(a)
-37, -9 (b)
-10, 0 (c)
it has 2 min. and 1 max. values
(d)
it has 2 max. and 1 min. values
50
Punjab EDUSAT Society (PES)
MATHEMATICS
10.
11.
point
12.
What is the maximum value of (1/x)x ?
(a)
(e)1/e (b)
(1/e)e (c)
e-e
(d)
none of these
25
On the interval [0, 1] the function x (1 – x)75 takes its maximum value at the
(a)
0
sin x + cos x
(b)
¼
(c)
½
(a)
(b)
≥2
(c)
≤ 2 (d)
≤2
(d)
1/3
≤ 1
2
Minimum value of (3 sin θ - 4 cos θ + 7) is
(a)
1/12 (b)
5/12 (c)
7/12 (d)
1/6
14.
The minimum value of f(a) = (2a2 – 3) + 2(3 – a) + 4
(a)
15/2 (b)
11/2 (c)
-13/2 (d)
13/2
15.
The number of values of x where f(x) = cos x + cos 2 x attains its maximum
value is
(a)
1
(b)
0
(c)
2
(d)
infinite.
2
16.
If y = a ln x + bx + x has its extremum values at x = -1 and x = 2, then :
(a)
a = 2, b = -1 (b)
a = 2, b = -1/2 (c)
a = -2, b = ½ (d)
none
-1
13.
17.
18.
π
π
The maximum value of cos 2 − x − cos 2 + x is
3
3
3
1
3
3
(a)
−
(b)
(c)
(d)
2
2
2
2
3
2
If f ( x) = 2 x − 21x + 36 x − 30, then which one of the following is correct ?
(a)
f ( x) has minimum at x = 1
(b)
f ( x) has maximum at x =
6
(c)
f ( x) has maximum at x = 1
(d)
f ( x) has no maxima or
minima
19.
The maximum distance from origin of a point on the curve x = a sin t – b
at
at
sin ,
y = a cos t − b cos , both a, b > 0 is
b
b
(a)
a – b (b)
a + b (c)
a2 + b2
(d)
a2 − b2
20.
If the function f(x) = 2x3 – 9ax2 + 12a2x + 1, where a > 0, attains its maximum
and minimum at p
and q respectively such that p2 = q, then a equals
(a)
½
(b)
3
(c)
1
(d)
2
21. The positive number x when added to its inverse gives the minimum value of
the sum at x equal to
(a)
-2
(b)
2
(c)
1
(d)
-1
x2 y 2
22.
Area of the greatest rectangle that can be inscribed in the ellipse 2 + 2 = 1
a
b
is
(a)
2ab
(b)
ab
(c)
ab (d)
a/b
51
Punjab EDUSAT Society (PES)
MATHEMATICS
x 2
+ has a local minimum at
2 x
(a)
x = 0 (b)
x = 1 (c)
x = 2 (d)
x = -2
2
3x + 9 x + 17
24.
If x is real, the maximum value of
is
3x 2 + 9 x + 7
(a)
1
(b)
17/7 (c)
¼
(d)
41
25. A triangular park is enclosed on two sides by a fence and on the third side by a
straight river bank. The two sides having fence are of same length x. The
maximum area enclosed by the park is
23.
The function f ( x ) =
(a)
½ x2
πx2
(b)
(c)
3/2 x2 (d)
x3
8
26.
If f(x) = x3 + bx2 + cx + d and 0 < b2 < c, then (-∞, ∞)
(a)
f(x) is a strictly increasing function (b)
f(x) has a local maxima
(c)
f(x) is a strictly decreasing function (d)
f(x) is bounded
27.
The minimum area of triangle formed by the tangent to the ellipse
and coordinate
(a)
x2 y 2
+
=1
a 2 b2
axes is
ab
(a + b)
(b)
a 2 + b2
2
2
a 2 + ab + b 2
2
3
28.** f(x) is cubic polynomial which has local maximum at x = -1. If (2) = 18, f(1)
= -1, f(x) has local
minima at x = 0, then
(a) the distance between (-1, 2) and (a, f(a)) where x = -a is the point of local
minima is 2 5
(b)
f(x) is increasing for x ∈ [1, 2 5 ] (c)
f(x) has local minima at x
=1
(d)
the value of f(0) = 5
ex
, 0 ≤ x ≤1
x
x −1
29.** f ( x ) = 2 − e , 1 < x ≤ 2 and g ( x ) = ∫ f ( t ) dt , x ∈ [1, 3] then
0
x −e , 2 < x ≤ 3
(c)
(a)
(b)
(c)
(d)
g(x) has local maxima at x = 1 + ln 2 and local minima at x = e
f(x) has local maxima at x = 1 and local minima at x = 2
f(x) has no local maxima
(d)
no local minima
30.
If f(x) is a twice differentiable function such that f(a) = 0, f(b) = 2, f(c) = -1,
f(d) = 2, f(e) = 0.
Where a < b < c < d < e. then the minimum number of zeros of
2
g(x) = (f′(x)) + f″(x)f(x) in the
interval [a, e] is
(a)
5
(b)
6
(c)
7
(d)
8
52
Punjab EDUSAT Society (PES)
MATHEMATICS
ANSWERS(Application of Derivative-II)
a
3.
d
4.
b
5.
b
1.
c
2.
6.
c
7.
a
8.
b
9.
a
10.
a
11.
b
12.
c
13.
a
14.
d
15.
a
16.
b
17.
c
18.
c
19.
b
20.
d
21.
c
22.
a
23.
c
24.
d
25.
a
26.
a
27.
a
28.
b,d
29.
a,b
30.
b
53
Punjab EDUSAT Society (PES)
MATHEMATICS
54
Punjab EDUSAT Society (PES)
MATHEMATICS
Integral Calculus
SECTION - A
1.
Integrate the following functions
cos x + sin x
(ii) cos 2x
cos x − sin x
(i) cos 2 2x sin 3x sin 5x
(iii) sin 6 x
(iv)
1
1 + tan x
(ii)
sin 2x
sin x + cos 4 x
ex − 1
(v) x
e +1
2.
3.
(i)
1
1 + sin 2 x
(iii)
1
sin x + cot x
(v)
1
sin x + tan 2 x
4
(iv) 1 + sec x
2
(i) cot3x cosec4x
(ii) (ex − 1)1/2
1− x
(iii) tan −1
1+ x
1/ 2
(iv)
tan x
(v) sec3 x cosec2x
4.
(i)
1
2x − 3x + 5
(iii)
(v)
5.
(i)
(iii)
2
1
2x − x + 1
2
(ii)
(iv)
1
3x − 2x − 1
2
1
2 − 3x − 5x 2
2x + 1
3x − 2x + 1
2
5x − 11
2x 2 − 3x + 3
x2 + 2
x3 −1
(ii)
x2
x 2 + 7x + 10
(iv)
1
(x + 1) (x + 2)
2
55
Punjab EDUSAT Society (PES)
MATHEMATICS
6.
7.
(v)
1
x − x2
(i)
x2
(x + 1) 2 (x + 2)
4
(iii)
x
(1 + x)(1 + x 2 )
(v)
1
(x + 1) (x + 2)
(ii)
x2 + 2
(x 2 + 3)(x 2 + 4)
(iv)
2x + 1
3x − 2x + 1
2
2
(i) x2 sin3x
(ii) e5x cos
x2
(iii) e
(x + 2)2
(iv) e x
x
3
5x
1 + sin x
1 + cos x
1 − sin x
(v) e− x / 2
1 + cos x
1/ 2
8.
(i)
1
sin x + sin 2x
(ii)
3sin x − 4 cos x
(iii)
2 sin x + cos x
9.
(v)
tan x
1 + tan x + tan 2 x
(i)
∫ 1 − sin
1
4
x
dx
1/ 2
sin(x − α)
(iv)
sin(x + α)
(ii)
(iii) If 0 < a < 1 then find
10.
1
∫ (2sin x + 3cos x) dx
dx
∫ 1 − 2a cos x + a
x
1
(iii) ∫ x log 1 + dx (iv)
x
x 3 − 6x 2 + 10x − 2
∫ x 2 − 5x + 6 dx
x 4 + 4x 3 + 11x 2 + 12x + 8
∫ (x 2 + 2x + 3)2 (x + 1) dx
1
11.
2
1 + sin 2x
(ii) ∫ e 2x
dx
1 + cos 2x
(i) ∫ ea sin bx dx
(v)
2 − sin x
2 + sin x
(i) ∫ x (1 + 3 − x )
3
0
2a
4 1/ 4
dx
(ii)
∫ (2ax − x
2 1/ 2
)
dx
0
56
Punjab EDUSAT Society (PES)
MATHEMATICS
π
1
(iii) ∫ x 2 tan −1 x dx
(iv)
0
π/ 2
∫
(v)
0
π/ 4
12.
∫
(i)
0
sin x + cos x
dx
sin 4x + cos 4x
dx
2 + tan x
π/ 4
(ii)
dx
∫0 x + (a 2 − x 2 )1/ 2 (iv)
2
13.
(i) ∫ | x 2 + 2x − 3 | dx (ii)
0
(iii) ∫ [x ]dx
4
π/ 2
(iv)
∫
0
x sin x
dx
cos3 x
1.5
∫ (x
2
) dx
∫
∫ | x |dx
−1
1
log sin x dx
(ii) ∫ cot −1 (1 − x + x 2 ) dx
0
0
π/ 2
∫
(iii)
0
dx
1 2
1 + sin x
6
∞
15.
π/ 4
1
0
(i)
sin x + cos x
dx
9 + 16sin x2x
0
a
14.
∫
0
a
(iii)
sin 4x
dx
sin x
0
∫
x log x
(i) ∫
dx
(1 + x 2 ) 2
0
1
(iv) ∫ (1 − x 2 )3/ 2 dx
0
∞
x2
(ii) ∫
dx
1+ x4
0
π
π
1− x2
(iii) ∫
dx (iv) ∫ log(1 + cos x)dx
1 − x 2 sin 2 α
0
0
π/ 2
16.
∫
(i)
0
1
− sin x dx
2
π
(ii) ∫ | cos x |dx
0
π/3
(iii)
π/2
∫ | tan x − cot x |dx
π/6
π
17.
(i)
∫
0
1
− cos x dx
2
∫
0
1
− sin −1 x dx
4
π
(iii) ∫ || sin x | − | cos x ||dx
0
(cos x − cos3 x) dx
−π / 2
π/ 2
(ii)
∫
(iv)
π/ 2
(iv)
∫ | sin x − cos x |dx
0
57
Punjab EDUSAT Society (PES)
MATHEMATICS
3
18.
1.5
(i) ∫ [x]dx
(ii)
0
(iv) ∫ [x 2 ]dx
0
2
2
(i) ∫ [x − x 2 ]dx
(ii) ∫ [x 2 − x + 1]dx
0
0
3π
(iii)
]dx
2
∫ x − [x]dx
−1
19.
2
0
1
(iii)
∫ [x
5π / 2
∫ [sin x]dx
(iv)
0
∫
[tan x]dx
0
20.
1
t 4 sin + t 2
t
If f (x) = ∫
dt , then prove that lim f ′(x) = −1
x →−∞
1
+
|
t |3
1
21.
Prove that
x
1
π(a 2 + b 2 )
dx
=
∫ a 2 sin 2 x + b2 cos2 x
4a 3 b3
tan x
22.
∫
Prove that
e −1
cot x
1
dt
dt + ∫
=1
2
1+ t
1+ t2
e −1
x
log t
1 1
dt x > 0 . Prove that f (x) + f = (log x)2
t +1
x 2
1
23.
f (x) = ∫
24.
If
25.
If Im, n = ∫ cos m x cos n x dx show that (m + n)I m,n = cos m x sin nx + m I m−1, n −1
26.
If
27.
If f : R → R is differentiable and f(1) = 4 then prove that lim
∫
log(x + 1 + x 2 )
1+ x2
dx = fog(x) + cos x
4e x + 6e − x
2x
∫ 9e x − 4e− x dx = Ax + Blog(9e − 4) + c , then find A, B and C
f (x)
x →1
∫
−2t dt = 8f ′(1)
4
x
28.
If g(x) = ∫ cos 4 x dx then show that g(x + π) = g(x) + g(π)
0
1
29.
Evaluate
∫ (x − [x]) dx
−1
π/4
30.
Evaluate
∫ log(1 + tan x) dx
0
58
Punjab EDUSAT Society (PES)
MATHEMATICS
x4
∫ 1 − x4
−1/ 3
2/ 3
31.
Evaluate
32.
(i)
3
2
−π
3 −1
12
π
3+
6
16.
(i)
17.
(i)
18.
(i) 3
19.
(i)
24.
dx
3π / 4
x dx
5−x + x
∫
−1 2x
cos
2
1+ x
(ii)
3π / 4
φ dφ
∫π / 4 1 + sin φ
(iii)
dx
1 + cos x
π/4
∫
Answers
4
4
(ii) 2π (iii) log
(iv)
3
3
1
1
(ii) sin −1 − 1
(iii) 4( 2 − 1)
2
4
(ii) 2 3
(iii) −1
5 −5
5− 5
(ii)
2
2
2
x
f (x) = g(x) = log(x + 1 + x 2 )
2
(iii) 0
(iv) 2 2 − 2
(iv) 5 − 2 − 3
(iv)
π
4
SECTION – B
1.
2.
3.
4.
dx
∫1+ e
x
=
(a) log(1 + e x ) + c
(b) − log(1 + e− x ) + c
(c) − log(1 − e x ) + c
(d) log(e x + e −2x ) + c
∫
1 − sin 2xdx =
π
, where x ∈ 0,
4
(a) − sin x + cos x + c
(b) sin x − cos x + c
(c) tan x + sec x + c
(d) sin x + cos x + c
1 + cos 2 x
∫ sin 2 x dx =
(a) cot x − 2x + c
(b) −2cot x + 2x + c
(c) −2cot x − x + c
(d) −2cot x + x + c
∫e
x
1
dx =
+ e− x
(a) tan −1 (e x ) + c
(b) tan −1 (e − x ) + c
(c) log(e x + e − x ) + c
(d) log(e x − e− x ) + c
59
Punjab EDUSAT Society (PES)
MATHEMATICS
5.
6.
dx
∫ x(1 + log x) =
(a) log(1 + log x)
(b) log(log(1 + logx))
(c) logx + log(logx)
(d) none of these
∫
x dx
4 − x4
=
x2
2
(b)
x2
1
cos −1
2
2
x2
2
(d)
1 −1 x 2
sin
2
2
(a) − cos −1
(c) − sin −1
7.
To find the value of
∫
(a) logx = t
1 + log x
dx , a proper substitution is
x
(b) 1 + logx = t
(c)
1
=t
x
(d) none
of
these
8.
9.
∫ x sec
2
x dx =
(a) tan x 2
(b) tan 2 x
(c) x tanx – log snx
(d) x tanx + log cosx
∫ x sin x dx = −x cos x + A , then A =
(a) sinx + c
these
10.
sin 2x
∫ 1 + sin
2
x
(b) cosx + c
π/4
11.
sec x dx
0
(a)
2
x
(d) tan −1 (sin x) + c
=
1
π
log( 2 + 1) +
3
2 2
of
(b) log(1 + sin 2 x) + c
1
log(1 + sin 2 x) + c
2
∫ 1 + 2sin
(d) none
dx =
(a) log sin 2x + c
(c)
(c) c
π
(c) 3 log( 2 + 1) −
2 2
(b)
1
π
log( 2 + 1) −
3
2 2
π
(d) 3 log( 2 + 1) +
2 2
60
Punjab EDUSAT Society (PES)
MATHEMATICS
π
12.
∫e
cos 2 x
cos5 3x dx =
0
(a) 1
these
π/2
13.
∫ 1+
0
1
tan x
(c) 0
(d) none
(b) π / 4
(c) π / 6
(d) 1
of
dx =
(a) π / 2
14.
(b) -1
The area bounded by curve y2 = x, line y = 4 and y-axis is
(a)
16
3
(b)
64
3
(c)
(d) none
2
of
these
2
15.
∫ log x dx =
1
1
(a) log
e
16.
(b) log4
(c) log
πx
1
+ B, f ′ = 2 and
2
2
If f (x) = Asin
1
∫ f | x |dx =
0
4
e
(d) log2
2A
, the values of A and B
π
are respectively
(a)
π
π
and
2
3
(b)
2
3
and
π
π
(c)
4
and 0
π
(d) 0 and
4
π
2π
17.
The value of
∫ [2sin x]dx , where [ ] is greatest integer function, is
π
(a) −π
(b) −2π
(c) −
5π
3
(d)
5π
3
2
18.
For evaluating the integral
∫ (px
2
+ qx + s) dx , the value the following constants
−2
are
(a) p
(b) q
(c) s
(d) p and s
(c) 6/15
(d) 8/15
π/2
19.
∫
sin 2 x cos 2 x(sin x + cos x)dx =
−π / 2
(a) 2/15
20.
(b) 4/15
The area bounded by the curve y = sinx, y = 0, x = 0 and x = π / 2 is
(a) π
(b) 2π
(c) 1
(d) 2
61
Punjab EDUSAT Society (PES)
MATHEMATICS
SECTION – C
1.
Let f : R → R and g : R → R be continuous function. Then the value of the
integral
π/2
∫ [f (x) + f (−x)][g(x) − g(− x)] dx
is
−π / 2
(a) π
(b) 1
(c) -1
(d) 0
(b) 2
(c) 4
(d) none
of
(c) 1 + e −1
(d) none
of
(c) 2
(d) 3
(c) 3 log2
(d) 4 log2
x2
∫ cos t
2.
2
dt
0
lim
x →0
x sin x
=
(a) 1
these
1
3.
The value of the definite integral ∫ (1 + e − x )dx is
2
0
(a) −1
these
(b) 2
x
4.
Solution of the equation
∫
x + 1 dx = 0 is
3
(a) 0
(b) 1
2
5.
The value of
dx
∫ 1+ | x − 1|
−2
(a) log2
π/4
6.
∫
0
(a)
(b) 2 log2
(sin θ + cos θ)
equals
(9 + 16sin 2θ)
1
10log 2
(b)
1
20log 5
(c)
1
log 3
20
(d)
(b)
1
(log x) 2
2
(c)
log x 2
2
(d) none
1
30log 7
x
7.
log(x 2 )
∫1 x dx
(a) (logx)2
of
these
62
Punjab EDUSAT Society (PES)
MATHEMATICS
π
8.
The value of integral ∫ x f (sin x) dx is
0
π
(b) π ∫ f (sin x) dx
(a) 0
π
(c)
π
f (sin x)dx
2 ∫0
(d) none
of
(c)
log x 2
2
(d) none
of
0
these
x
9.
log(x 2 )
∫1 x dx
(a) (log x) 2
(b)
1
(log x) 2
2
these
10.
Let f : R → R be differentiable function and f(1) = 4. Then the value of
f (x)
lim
x →t
∫
4
2t
dt is
x −1
(a) 8f ′(1)
11.
(b) 2
2
The area of the ellipse
(d) f ′(1)
(c) 2 2
(d) 4
(c) π(a + b)
(d)
x 2 y2
+
= 1 is
a 2 b2
(b)
(a) 6π
13.
(c) 2f ′(1)
The area enclosed within the curve |x| + |y| = 1 is
(a)
12.
(b) 4f ′(1)
π(a 2 + b 2 )
4
πab
4
Area of the region bounded by the curve y = tanx, tangent drawn to the curve
at x = π / 4 and the x-axis is
(b) log 2 +
(a) log 2
1
4
(c) log 2 −
1
4
(d)
1
4
π
14.
The value of
cos 2 x
∫ 1+ ax
−π
(a) π
(b) aπ
e2
15.
The value of the integral
∫
e−1
(a) 3/2
16.
(c) π / 2
(d) 2π
(c) 3
(d) 5
log e x
dx is
x
(b) 5/2
The area bounded by the curves y = f(x), the x-axis and the ordinates x = 1 and
x = b is (b − 1)sin(3b + 4) . Then f(x) is
(a) (x − 1) cos(3x + 4)
(b) sin(3x + 4)
63
Punjab EDUSAT Society (PES)
MATHEMATICS
(c) sin(3x + 4) + 3(x − 1)cos(3x + 4)
(d) none of these
x
17.
Let f : (0, ∞) → R and F(x) = ∫ f (t) dt . If F(x2) = x2(1 + x), then f(4) equals
1
(a) 5/4
(b) 7
(c) 4
x
18.
Let g(x) = ∫ f (t) dt where f is such that
0
(d) 2
1
1
≤ f (t) ≤ 1 for t ∈ [0, 1] and 0 ≤ f (t) ≤
2
2
for t ∈ (1, 2] . Then g(2) satisfies the inequality
3
2
(a) − ≤ g(2) ≤
(c)
1
2
(b) 0 ≤ g(2) < 2
3
5
< g(2) ≤
2
2
(d) 2 < g(2) < 4
x
19.
If g(x) = ∫ cos 4 t dt , then g(x + π) equals
0
(a) g(x) + g( π)
(b) g(x) − g(π)
(c) g(x)g(π)
(d)
g(x)
g(π)
3π / 2
20.
If for a real number y, [y] is the greatest integer ≤ y , then
∫
[2sin x]dx is
π/2
(a) −π
21.
ecos x sin x for | x | ≤ 2
If f (x) =
, then
otherwise
2
(a) 0
22.
(c) −π / 2
(b) 0
(b) 1
(d) π / 2
3
∫ f (x)dx =
−2
(c) 2
(d) 3
2
tan −1 1 + x + x
e
∫ 1 + x 2 dx
−1
xe tan x
(a)
(1 + x 2 ) 2
−1
(b) xe
tan −1 x
xe tan x
(c)
1 + x2
(d) none
of
these
α
23.
The value of the integral
x dx
∫ 1 + cos α sin x , 0 < α < π is
0
(a)
πα
sin α
(b)
πα
1 + sin α
(c)
πα
cos α
(d)
πα
1 + cos α
64
Punjab EDUSAT Society (PES)
MATHEMATICS
24.
If f (x) = ae 2x + be x + cx , satisfies the conditions f(0) = -1
f ′(log 2) = 31 ,
log 4
∫ [f (x) − x ] dx =
0
39
, then
2
(a) a = 5, b = 6, c = 6
(b) a = 5, b = − 6, c = 3
(c) a = − 5, b = 6, c = 3
(d) none of these
sin x + x 2
∫0 3− | x | dx is equal to
1
25.
sin x + x 2
∫0 3− | x | dx
1
(a)
1
1
(b) 0
(c)
x2
∫0 3− | x | dx
(d)
sin x
∫ 3− | x | dx
0
k
26.
Let f be a positive function, let I1 =
∫
k
x f (x(1 − x)) dx , I 2 =
1− k
where 2k – 1 > 0. Then
(a) 2
27.
∫
x
a
x
∫ f (x(1 − x)) dx ,
1− k
I1
is
I2
(b) k
(c) 1/2
(d) 1
dx =
(a) a x log e a + c
(b) 2a x log e a + c
(c) 2a x log10 a + c
(d) 2a x log a e + c
1
28.
The value of the definite integral ∫ (1 + e − x ) dx is
2
0
(a) -1
these
29.
∫5
55
x
(c) 1 + e −1
(b) 2
(d) none
of
(d) none
of
⋅ 55 ⋅ 5x dx is equal to
x
5x
x
55
(a)
+c
(log 5)3
x
(b) 5 (log 5) + c
55
3
55
(c)
+c
(log 5)3
these
30.
The indefinite integral ∫ ( tan x + cot x ) dx is
65
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
(c)
cot x + 1
tan −1
2
2 tan x
tan x
tan x + 1
2 tan −1
2 tan x
(b)
(d)
tan x − 1
2 tan −1
2 tan x
cot x − 1
tan −1
2
2cot x
tan x
66
Punjab EDUSAT Society (PES)
MATHEMATICS
Complex numbers
PROBLEM SET ON COMPLEX NUMBERS
1.
The locus of the center of a circle which touches the circle z – z1= a and z
– z2= b externally (z, z1 & z2 are complex numbers) will be
(a)
an ellipse
(b)* a hyperbola (c)
a circle
(d)
none of these
If z and w are two non – zero complex numbers such that zw= 1 and arg
2.
(z) – arg (w) = π/2, then zw is equal to
(a)* -i
(b)
1
(c)
-1
(d)
I
3.
Let z1 and z2 be two roots of the equation z2 + az + b = 0, z being complex.
Further, assume that the origin, z1 and z2 form an equilateral triangle. Then
(b)
a2 = b
(c)
a2 = 2b
(d)* a2 =
(a)
a2 = 4b
3b
4.
1+ i
x
If
= 1 n is a positive integer then
1− i
(a)
x = 2n + 1
(b)* x = 4n
(c)
x = 2n
(d)
x=
4n + 1
5.
Let z, w be complex numbers such that z + iw = 0 and arg zw = π. Then arg z
equals
(a)
5π/4
(b)
π/2
(c)* 3π/4 (d)
π/4
6.
x y
a+ b
If z = x – iy and z1/3 = a + ib, then 2 2 is equal to
a +b
(
(a)*
-2
(b)
-1
)
(c)
If z2 - 1= z2 + 1, then z lies on
(a)
an ellipse
(b)
the imaginary axis
real axis
2
(d)
1
(c)*
a circle (d)
7.
the
8.
If the cube roots of unity are 1, ω, ω2 then the roots of the equation (x – 1)3 + 8
= 0 are
(a)
-1, -1 + 2ω, -1 - 2ω2
(b)
-1, -1, -1
2
(c)* -1, 1 - 2ω, 1 - 2ω
(d)
-1, 1 + 2ω, 1 + 2ω2
9.
If z1 and z2 are two non-zero complex numbers such that z1 + z2=
z1+z2, then
67
Punjab EDUSAT Society (PES)
MATHEMATICS
arg z1 – arg z2 is equal to
(a)
π/2
(b)
10.
If ω =
(a)
parabola
z
1
z− i
3
an ellipse
The value of
(b)
∑ sin
k =1
(a)
12.
(d)
-π/2
(c)*
0
(c)*
a straight line (d)
(c)
i
and ω = 1, then z lies on
10
11.
-π
-1
a circle
a
2kπ
2 kπ
+ i cos
is
11
11
(b)*
-i
(d)
1
(d)
54
If z2 + z + 1 = 0, where z is a complex number, then the value of
2
2
2
2
1 2 1 3 1
6 1
z + + z + 2 + z + 3 + ...... + z + 6 is
z
z
z
z
(a)
6
(b)*
12
(c)
18
13.
If z + 4 ≤ 3, then the maximum value of z + 1 is
(a)
10
(b)* 6
(c)
0
(d)
4
14.
The conjugate of a complex number is
(a)*
15.
−1
i +1
(b)
1
i −1
(c)
1
is
1 − cos θ + i sin θ
1
1
(b)*
1 − cosθ
2
1
. Then that complex number is
i −1
−1
1
(d)
i −1
i +1
The real part of
(a)
(c)
tan θ
2
(d)
2
16.
The points representing complex number z for which z - 3= z - 5 lie on
the locus given by
(a)
circle
(b)
ellipse
(c)* straight line (d)
none of these
17.
z - i < z + i represents the region
(a)
Re(z) > 0
(b)
Re (z) < 0
(c)*
Im (z) > 0
(d)
Im
(z) < 0
68
Punjab EDUSAT Society (PES)
MATHEMATICS
18.
If x = 2 + 5i (where i2 = -1), then the value of (x3 – 5x2 + 33x – 29) is equal to
(a)
6
(b)
8
(c)* 0
(d)
12
19.
If 1, ω, ω2, ……….., ωn-1 are n roots of unity, then (1 - ω)(1 - ω2) ……..(1 ω n −1 ) is equal to
(a)* n
(b)
1
(c)
0
(d)
n2
20.
21.
The maximum number of real of the equation x2n – 1 = 0 is
(a)* 2
(b)
3
(c)
n
none of these
If the imaginary part of
(d)
2z + 1
is -2, then the locus of the point represented by
iz + 1
z, is a
(a)
circle
none of these
22.
23.
(b)*
If
parabola
(d)
(d)
1 − ix
2
2
= a + ib , then a + b =
1 + ix
Let a = cos
(b)
-1
(c)
0
(d)
2π
2π
+ i sin
, A = a + a 2 + a 4 and B = a3 + a 5 + a 6 , then A and B are
7
7
roots of the equation
(a)
x2 – x + 2 = 0 (b)
none of these
25.
(c)
z −1 π
The locus of the point z satisfying the condition arg
= is
z +1 3
(a)
parabola
(b)* circle
(c)
pair of straight line
none
(a)* 1
none of these
24.
straight line
x2 – x – 2 = 0 (c)*
x2 + x + 2 = 0 (d)
The complex number z = x + iy which satisfy the equation
z − 5i
= 1 lies on
z + 5i
(a)*
the axis of x
(b)
the straight line y
(c)
the circle passing through the origin
(d)
none of these
=5
69
Punjab EDUSAT Society (PES)
MATHEMATICS
26.
Let α and β be the roots of the equation x2 + x + 1 = 0 the equation whose
roots are α19, β7 is
(a)
x2 – x – 1 = 0 (b)
x2 – x + 1 = 0 (c)
x2 + x – 1 = 0 (d)* x2
+x+1=0
27.
If the complex numbers iz, z and z + iz represent the three vertices of a
triangle, then the area of
the triangle is
1 2
z
2
(a)*
z −1
28.
(b)
z2
(c)
1
z −1
2
(d)
(c)
-2
(d)
2
The product of cube roots of -1 is equal to
(a)* -1
(b)
0
4
29.
If magnitude of a complex number 4 – 3i is tripled and rotated by an angle π
anticlockwise then
resulting complex number would be
(a)* -12 + 9i
(b)
12 + 9i
(c)
7 – 6i
(d)
7 + 6i
30.
The complex number z1, z2, z3 satisfying
(a)*
(c)
equilateral
acute angled triangle
z1 − z3 1 − i 3
then triangle is
=
z 2 − z3
2
(b)
(d)
right angled triangle
obtuse angled isosceles
31.
If z – 2 – 3i + z + 2 – 6i = 4 where i = −1 then locus of P(z) is
(a)
an ellipse
(b)* φ
(c)
segment joining the point 2 + 3i, -2 + 6i
(d) none
32.
If lm
= −4 , then locus of z is
2z + 1
(a)* ellipse
(b)
parabola
circle
33.
z −1
If
(a)
34.
If
(c)
straight line
(d)
z +i
= 3 , then radius of the circle is
z −i
2
21
1
(b)
21
(c)*
(d)
3
21
b
d
3 + i = ( a + ib )( c + id ) , then tan −1 + tan −1 has the value
a
c
(a)
π/3 + 2nπ, n ∈ I
(c)
nπ - π/3, n ∈ I
(b)*
(d)
nπ + π/6, n ∈ I
2nπ - π/3 , n ∈ I
70
Punjab EDUSAT Society (PES)
MATHEMATICS
35.
The amplitude of sin
(a)
36.
2π
5
π
π
+ i 1 − cos is
5
5
(b)
π
15
(c)*
π
10
(d)
z1 + z2
(d)
π
5
1
1
( z1 + z2 ) + z1 z2 + ( z1 + z2 ) − z1 z2 is equal to
2
2
(a)
z1 + z2
(b)
z1 − z2
(c)*
z1 − z2
37.
a + bω + cω2 a + bω + cω2
+
The value of
will be
b + cω + aω2 c + aω + bω2
(a)
1
(b)*
-1
(c)
2
(d)
-2
71
Punjab EDUSAT Society (PES)
MATHEMATICS
Equations, Inequations & Inequalities
1.
The number of values of k for which the equation x2 – 3x + k = 0 has two
distinct roots lying in the interval (0, 1) is
(a)
three
(b)
two
(c)
infinite
(d)* no value of k will satisfy
2.
If α, β are the roots of the equation ax2 + bx + c = 0, then the value of α3 + β3
is
3abc + b3
a3
(a)
(
− 3abc + b3
a
(b)
)
a 3 + b3
3abc
(c)*
3abc − b3
a3
(d)
3
3.
If the ratio of the roots of the equation x2 + bx + c = 0 is as that of x2 + qx + r
= 0, then
(a)
r2b = qc2
(b)
r2c = qb2
(c)
c2r = q2b
(d)* b2r
2
=q c
4.
If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the
sum of the
square of their reciprocals, then
(a)* ab2, ca2, bc2 are in A.P
(b)
a2b, c2a, b2c are in
A.P
(c)
ab2, ca2, bc2 are in G.P
(d)
none of these
5.
If f(x) = 2x3 + mx2 – 13x + n and 2, 3 are roots of the equation f(x) = 0, the
values of m and n are
(a)
5, -30
(b)* -5, 30
(c)
5, 30
(d)
none of these
6.
If α , β
are the roots of ax 2 + bx + c = 0, and
1 1
, are the roots of
α β
(a)
cx + ax + b = 0
(b)
bx + ax + c = 0
(c)*
cx + bx + a = 0
(d)
ax 2 + cx + b = 0
2
2
2
7.
If the equation (λ − 1) x 2 + (λ + 1) x + (λ − 1) = 0 has real roots , then λ can have
any value in
the interval
(a)
(1/ 3,3)
(b) (−3,3)
(c) (0, 3)
(d)* (1/ 3, 2)
8.
Let |x| and [x] denote the fractional integral part of a real number x
respectively . Then solutions of 4|x| = x+[x] are
(a)
2
± ,0
3
4
3
(b) ± ,0
(c)*
0,
5
3
(d)
±2, 0
9.
The value of ‘a’ for which the equations x3+ax+1=0 and x4 + ax + 1 = 0
have a common roots is
(a)
2
(b)* -2
(c)
0
(d) none of these
72
Punjab EDUSAT Society (PES)
MATHEMATICS
10.
The roots of the equation x 2 − x − 6 = x + 2 are
(a)
−2,1, 4
(b)
0, 2, 4
(c)
0,1,4
(d)* −2, 2, 4
11.
If α and β are the roots of the equation x + px + 1= 0 and ν and δ are the
root of
x 2 + qx + 1= 0, value of (α − ν ) ( β − ν )(α + δ )( β + δ ) is
(a)
p2-q2
(b)* q2-p2
(c)
p2
(d)
q2
12.
If α and β are the roots of the equation ax 2 + bx + c = 0 then
(1 + α + α 2 ) (1 + β 2 + β 2 ) =
(a)
0
(b)* positive
(c)
negative
(d)
none of these
2
13.
α, β be the root of x2 – 3x + a = 0 and v, δ are the root of the x2 – 12x + b = 0
and numbers α, β,
v, δ (in order) from an increasing G.P., then
(a)
a = 3, b = 12 (b)
a = 12, b = 3 (c)* a = 2, b = 32 (d)
a=
4, b = 16
14.
(
log x 2 − 4 x + 5
The real roots of the equation 7 7
(a)
1 and 2
(b)* 2 and 3
) = x −1
(c)
3 and 4
(d)
4
(d)
4
and 5
15.
The number of real solutions of 2sin (ex) = 5x + 5-x in [0, 1] is
(a)* 0
(b)
1
(c)
2
16.
The value of ‘a’ for which one root of the quadratic equation (a2 – 5a + 3)x2 +
(3a – 1)x + 2 = 0 is
twice as large as the other is
(a)
- 1/3
(b)* 2/3
(c)
- 2/3
(d)
1/3
17.
The number of real solutions of the equation x2 - 3x+ 2 = 0 is
(a)
3
(b)
2
(c)* 4
(d)
1
18.
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these
numbers are the
roots of the quadratic equation
2
(b)* x2 – 18x + 16 = 0
(a)
x – 18x – 16 = 0
(c)
x2 + 18x – 16 = 0
(d)
x2 + 18x + 16 = 0
2
19.
If (1 – p) is a root of quadratic equation x + px + (1 – p) = 0 then its roots are
(a)
-1, 2
(b)
-1, 1
(c)* 0, -1
(d)
0, 1
20.
If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q
= 0 has equal roots, then the value of ‘q’ is
(a)
4
(b)
12
(c)
3
(d)*
49/4
21.
The value of α for which the sum of the squares of the roots of the equation
x2 – (α - 2)x - α - 1 = 0 assume the least value is
(a)* 1
(b)
0
(c)
3
(d)
2
73
Punjab EDUSAT Society (PES)
MATHEMATICS
22.
If the roots of the equation x2 – bx + c = 0 be two consecutive integers, then b2
– 4c equals
(a)
-2
(b)
3
(c)
2
(d)* 1
23.
If both the roots of the quadratic equation x2 – 2kx + k2 + k – 5 = 0 are less
than 5, then k lies in the interval
(a)
(5, 6]
(b)
(6, ∞)
(c)* (-∞, 4)
(d)
[4,
5]
24.
All the values of m for which both roots of the equation x2 – 2mx + m2 – 1 = 0
are greater than -2
but less than 4, lie in the interval
(a)* -1 < m < 3
(b)
1<m<4
(c)
-2 < m < 0
(d)
m>
3
25.
If the roots of the quadratic equation x2 + px + q = 0 are tan 30° and tan 15°
respectively then
the value of 2 + q – p is
(a)
0
(b)
1
(c)
2
(d)* 3
If p and q are positive real numbers such that p 2 + q 2 = 1, then the maximum
26.
value of (p +q ) is
1
1
(a)
(b)
(c)*
2
(d)
2
2
2
27
A value of C for which the conclusion of Mean Value Theorem holds for the
function
f ( x) = log e x on the interval [1,3] is
1
log e 3
(b)
log 3 e (c)
log e 3
(d)* 2log 3 e
2
28.
Let A ( h, k), B (1, 1 ) and C (2, 1) be the vertices of a right angled triangle
with AC as its
hypotenuse. If the area of the triangle is 1, then the set of
values which ‘k’ can take is given by
(a)
{0 , 2}
(b)* {-1 , 3}
(c)
{-3 , -2}
(d)
{1 ,
3}
29.
If the difference between the roots of the equation x 2 + ax + 1 = 0 is less then
(a)
5, then the
set of possible values of a is
(a)
(−3, ∞ )
(b)
(3, ∞)
(c) (−∞ , −3)
(d)* ( - 3 , 3)
Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not
30.
all zero such that x = cy + bz , y = az + cx, and z = bx + ay. Then a 2 + b 2 + c 2 + 2abc is
equal to
(a)
-1
0
(b)*
1
(c)
2
(d)
74
Punjab EDUSAT Society (PES)
MATHEMATICS
PROBLEM SET ON PROBABILITY
1.
Two fair dice are tossed. Let X be the event that the first die show an even
number and Y be the event that second die show an odd number. The two events X
and Y are
(a)
mutually exclusive
(b)
independent and mutually
exclusive
(c)
dependent
(d)* none of these
2.
Two events A and B have probabilities 0.25 and 0.50 respectively. The
probability that both A
and B occur simultaneously is 0.14. Then the
probability that neither A nor B occurs is
(a)* 0.39
(b)
0.25 (c)
0.11
(d)
none of
these
3.
For a biased die the probabilities for the different faces to turn up are given
below:
Face
1
2
3
4
5
6
Probability 0.1
0.32
0.21
0.15
0.05
0.17
This die is tossed and you are told that either face 1 or face 2 has found up.
Then the probability that it is face 1 is
(a)*
5
21
(b)
8
21
(c)
11
21
(d)
13
21
4.
A determinant is chosen at random from the set of all determinants of order 2
with elements 0
or 1 only. The probability that the value of determinant chosen
is positive, is
(a)*
5.
3
16
(b)
5
16
(c)
7
16
(d)
9
16
If A and B are two events such that P(A) > 0, and P(B) ≠ 1, the P ( A / B ) is
equal to
(a)
1 – P(A/B)
(b)
(
1− A / B
)
(c)*
( )
P(B)
1− P ( A ∪ B)
P(B)
(d)
P A
6.
Fifteen coupons are numbered 1, 2, ……….., 15 respectively. Seven coupons
are selected at random time with replacement. The chance that the largest number
appearing on a selected
coupon is 9, is
(a)
9
16
6
(b)
8
15
7
(c)
3
5
7
(d)*
none of these
75
Punjab EDUSAT Society (PES)
MATHEMATICS
7.
If the letters of word “ASSASSIN” are written down at random in a row, the
probability that no
two S’s occur together, is
(a)* 1/14
(b)
1/35
(c)
1/7
(d)
none of these
8.
Three identical dice are rolled. The probability that the same number will
appear on each of
them is
(a)
1/6
(b)* 1/36
(c)
1/18
(d)
3/28
9.
If M and N are any two events, the probability that exactly one of them occurs
is
(a)* P(M) + P(N) – 2P(M ∩ N)
(b)
P(M) + P(N) – P(M ∩ N)
P ( M ) + P ( N ) − 2P ( M ∩ N )
(d)
P(M ∩ N ) − (M ∩ N )
(c)*
10.
A box contain 100 tickets numbered 1, 2, ……………, 100. Two tickets are
chosen at random. It is
given that the maximum number on the tickets is not
more than 10. Then the probability that
minimum on them is 5, is
(a)* 1/9
(b)
2/9
(c)
1/3
(d)
2/5
11.
12.
P(A ∪ B) = P(A ∩ B) is possible only when
(a)* P(A) = P(B)
(b)
(c)
P(A) < P(B)
(d)
If
P(A) > P(B)
none of these
1+ 3p 1− p
1− 2 p
,
and
are the probabilities of three mutually exclusive
3
4
2
events, then p lies in the interval
(a)
0≤p<½
(b)*
none of these
1/3 ≤ p ≤ ½
(c)
0 ≤ p ≤ 1/3
(d)
13.
A student appears for tests I, II and III. The student is successful if he passes
either in tests I and II or test I and III. The probability of the student passing in tests
I, II and III are p, q and ½
respectively. If the probability that the student is
successful is ½, then
(a)
p=q=1
(b)
p=q=½
(c)* p = 1, q = 0 (d)
p=
1, q = ½
(e)
none of these
14.
The probability that at least one of the events A and B occurs is 0.6. If A and
B occur
simultaneously with the probability 0.2, then P ( A ) + P ( B ) is
(a)
0.4
(b)
0.8
(c)*
15.** For two given events A and B, P(A ∩ B) is
(a)* Not less than P(A) + P(B) – 1
(b)*
P(B)
(c)* Equal to P(A) + P(B) – P(A ∪ B)
P(B) + P(A ∪ B)
1.2
(d)
1.4
Not greater than P(A) +
(d)
Equal to P(A) +
76
Punjab EDUSAT Society (PES)
MATHEMATICS
16.
Urn A contains 6 red and 4 black balls and urn B contains 4 red and 6 black
balls. One ball is
drawn at random from urn A and placed in urn B. Then one ball
is drawn at random from urn B
and placed in urn A. If one ball is now drawn at
random from urn A, the probability that it is found out be red is
(a)* 32/55
(b)
41/55
(c)
43/55
(d)
½
17.
In three throws of two dice, the probability of throwing doublets not more than
twice is
(a)
1/6
(b)
5/72
(c)* 215/216
(d)
none
18.
Given a throw of three unbiased dice show different faces, the probability that
at least one face
shows 6 is
(a)
5/6
(b)
5/18
(c)* ½
(d)
13/18
19.
Bag A contains 2 white and 2 red balls and another bag B contains 4 white
and 5 red balls. A
ball is drawn and is found to be red. The probability that it was
drawn from the bag B is
(a)
25/52
(b)
½
(c)* 10/19
(d)
5/19
20.
Let A and B be two events such that P(A) = 7/20, P(B) = 9/20, P(A ∪ B) =
11/20, P ( A ∪ A ) = 1 , then the value of P ( A ∪ B ) is equal to
(a)
¼
none of these
(b)*
¾
(c)
1/10
(d)
21.
A bag contains 5 brown and 4 white socks. A man pulls out two socks without
replacement. The
probability that they are of the same colour is
(a)
5/108
(b)
1/6
(c)
5/18
(d)* 4/9
22.
A six faced die is so biased that it is twice to show an even number as an odd
number when thrown. If it is thrown twice the probability that the sum of two
numbers thrown is even, is
(a)
4/9
(b)* 5/9
(c)
1/9
(d)
none of these
23.
In shuffling a pack of playing cards, four cards accidently dropped. The
probability that the missing cards should be one from each suit is
(a)
1
256
(b)
4
20825
(c)*
2197
20825
(d)
none of these
24.
The probability that a student is not a swimmer is 1/5. What is the probability
that out of 5 students, 4 are swimmer?
77
Punjab EDUSAT Society (PES)
MATHEMATICS
4
(a)*
5
4 1
C4
5 5
4
(b)
4
4 1
5 5
(c)
3
1 4
C1 × 5C4
5 5
(d)
none
25.
four persons are selected at random from a group of 3 men, 2 women and 4
children. What is
the chance that exactly 2 of them are children?
(a)
9/21
(b)
10/23
(c)
11/24
(d)*
10/21
26.
A person writes 4 letters and addresses 4 envelopes. If the letters are placed in
the envelopes at
random, what is the probability that all letters are not placed in
the right envelopes?
(a)
1/24
(b)
11/24
(c)
15/24
(d)*
23/24
27.
A coin is tossed 3 times by 2 persons. What is the chance that both get equal
number of heads?
(a)
3/8
(b)
1/9
(c)* 5/16
(d)
none of these
28.
Out of 15 tickets marked with the numbers from 1 to 15, three are drawn at
random. What is
the chance that the numbers on them are in A.P?
(a)* 7/65
(b)
9/15
(c)
13/261
(d)
none of these
29.
The probability of event A occurring is 0.5 and B occurring is 0.3. If A and B
are mutually exclusive events, then the probability of neither A nor B occurring is
(a)
0.6
(b)
0.5
(c)
0.7
(d)*
none of these
30.
If A and B are two independent events, the probability that only one of A or B
occur is
(a)* P(A) + P(B) – 2P(A ∩ B)
(b)
P(A) + P(B) – P(A ∩ B)
(c)
P(A) + P(B)
(d)
none of these
31.
If A and B are two independent events, the probability that both A and B occur
is 1/8 and the probability that neither of them occurs 3/8. The probability of the
occurrence of A is
(a)* ½
(b)
1/3
(c)
¾
(d)
1/5
32. Twelve balls are distributed among three boxes. The probability that the first
box contains 3 ball is
10
(a)*
110 2
9 3
10
(b)
9 2
110 3
12
(c)
C3 9
.2
123
(d)
12
C3
3
12
78
Punjab EDUSAT Society (PES)
MATHEMATICS
33.
A fair coin is tossed a fix number of times . If the probability of getting 4
heads equal the
probability of getting 7 heads .Then the probability of getting
2 heads is
(a)*
35
2048
3
4096
(b)
(c)
1
1024
(d)
none of these
34.
If A and B are two events that the value of the determinant chosen at random
from all the
determinants of order 2 with entries 0 or 1 only is positive or negative
respectively . Then
P ( A) > P( B ) (b)*
P ( A) ≤ P ( B ) (c)
P ( A) = P ( B ) = 1/ 2
(d)
(a)
none of these
35.
The probability of A = probability of B = probability of
1
C = , P( A ∩ B ∩ C ) = 0
4
1
P ( B ∩ C ) = 0, P ( A ∩ C ) = , and P ( A ∩ B ) = 0 , the probability that at least one
8
of the events A, B, or C will occur, is
5
37
3
(a)*
(b)
(c)
(d) 1
8
64
4
36.
India plays two matches each with West indies and Australia . In any match
the probabilities of India getting points 0 , 1 and 2 are 0.45 , 0.05 respectively .
Assuming that the outcomes are
Independent. The probability of India getting at least 7 points is .
(a)
1/7
(b)
.0675
(c)* .0875
(d)
none of these
37.
1
3
X is random variable with probability function P ( x ) = 3C x , x = 0,1, 2,3 the
2
value of
E ( X 2 + 2) is
(a)
38.
7
(b)
1
3
17/4
5
6
(c)*
5
(d)
4
1
2
If P( A ∩ B ) = , P( A ∪ B ) = and P ( A) = , then which one of the following is
correct?
(a)*
A and B are independent events
(b)
A and B are mutually
exclusive events
(c)
P( A) = P ( B )
(d)
P ( A) < P ( B )
39.
Bag A contains 4 green and 3 red balls and bag B contains 4 red and 3
green balls. One bag
is taken at random and a ball is drawn and noted it is green . The probability
that it comes
from bag B
(a)
2/7
(b)
2/3 (c)*
3/7 (d)
1/3
79
Punjab EDUSAT Society (PES)
MATHEMATICS
40.
For a party 8 guests are invited by a husband and his wife . They sit around
a circular table
for dinner . the probability that the husband and his wife sit together is :
(a)
2
7
2
9
(b)*
1
9
(c)
(d)
4
9
41.
One hundred identical coins, each with probability p of showing heads are
tossed once .
If 0 < p < 1 and the probability of head showing on 50 coins is equal to that of
head showing on
51 coins, the value of p is:
(a)
1
2
(b)*
51
101
49
101
(c)
(d) none of these
42.
An anti-aircraft gun can take a maximum of four shots at an enemy plane
moving away from it. The probabilities of hitting the plane at the first , second , third
and fourth shot are 0.4 , 0.3,0.2,
and 0.1 respectively. What is the probability that
the gun hits the plane
(a)
0.6977 (b) 0.6967
(c)* 0.6976 (d) none of these
43.
Five horses are in a race . Mr A selects two of the horse at random and bets on
them, the
probability that Mr A selected the winning horse is
(a)* 2/5
(b)
4/5
(c)
3/5
(d) 1/5
44.
The probability that A speaks truth is
4
3
, while this probability for B is
5
4
The probability
that they contradict each other when asked to
(a)
4/5 (b)
1/5 (c)* 7/20
(d) 3/20
45.
A random variables X has the probability distribution:
X:
1
2
3
4
5
6
7
8
P(X):
0.15
0.12
0.10
0.20
0.08
0.07
0.23
0.05
For the events E = { X is a prime number} and F = { X < 4}, the probability
P ( E ∪ F ) is :
(a)
0.50
(b) * 0.77 (c)
0.35
(d)
0.87
46.
The mean and the variance of a binomial distribution are 4 and 2 respectively.
Then the
probability of 2 successes is
(a)*
28
256
219
256
(b)
128
256
(c)
37
256
(d)
47.
Three houses are available in a locality .Three persons apply for the houses.
Each applies for
one house without consulting others. The probability that all
the three apply for the same house is.
2
9
(a)
1
9
(b)*
8
9
(c)
(d)
7
9
48.
A random variable X has Poisson distribution with mean 2. Then P( X > 1.5)
equals
(a)
2
e2
(b)
0
(c)*
1−
3
e2
(d)
3
e2
80
Punjab EDUSAT Society (PES)
MATHEMATICS
49.
1
6
Let A and B be two events such that P ( A ∪ B ) = , P ( A ∩ B ) =
1
P ( A ) = , where A
4
1
and
4
stands for complement of event A. Then events A and B are
(a)
equally likely and mutually exclusive(b)
independent
(c)* independent but not equally likely (d)
independent.
equally likely but not
mutually exclusive and
50.
At a telephone enquiry system the number of phone calls regarding relevant
enquiry follow
Poisson distribution with an average of 5 phone calls during 10minute time intervals. The probability that there is at the most one phone call
during a 10-minute time period is
(a)
6
55
(b)*
6
e5
(c)
6
5e
(d)
None of these
81
Punjab EDUSAT Society (PES)
MATHEMATICS
TRIGONOMETRY
1.
2.
If tan-1(2x) + tan-1(3x) = π/4, then x =
(a)
½
(b)
1/3
(c)
1/6
4.
1 + x2
(b)
x
(c)
tan 10° + tan 35° + tan 10° . tan 35° =
(a)
0
(b)
½
(c)
-1
(1 + x )
−3
(d)
1
2
2
(d)
(1 + x )
2
The general value of θ satisfying the equation sin θ = - ½ and tan θ = 1
(a)
(c)
5.
1/10
The value of sin (cot-1 x) =
(a)
3.
(d)
If
equal to
(a)
nπ +
π
,n∈ I
(b)
6
7π
2nπ +
,n∈I
6
(d)
−1
3
2
is
n 7π
nπ + ( −1)
, n ∈ I
6
11π
2nπ +
,n∈ I
6
cos A cos B 1 π
π
=
= , − < A < 0, − < B < 0, the value of 2 sin A + 4 sin B is
3
4
5 2
2
4
(b)
2
(c)
-4
(d)
0
6.
From the top of a light house the angle of depression of a boat is 15. If the
light house is 60 m high and its base is sea-level, the distance of the boat from the
light house is
(a)
3 −1
60
3 +1
(b)
3 +1
60
3 −1
2
(c)
3 −1
60
3 +1
(d)
none of these
7.
The value of log3 tan 1° + log3 tan 2° + ………… + log3 tan 89° is
(a)
3
(b)
1
(c)
2
(d)
0
8.
If in a triangle ABC, c = 2a cos B, then the triangle is a/an
(a)
simple triangle
(b)
isosceles triangle
(c)
equilateral triangle
(d)
right-angle triangle
9.
AB is a vertical diameter of circle and PQ is a diameter at an angle θ to AB.
The value of θ, so
that the time of sliding down PQ any be 2 times that of
sliding down AB, is
(a)
π/6
(b)
π/4
(c)
π/3
(d)
none of these
82
Punjab EDUSAT Society (PES)
MATHEMATICS
10.
A flagstaff, 10 meters high, stands at the centre of an equilateral triangle,
which is horizontal. On
the top of the flagstaff each side subtends an angle of
60°. The length of the each side is
6 3 (b)
4 6 (c)
5 6 (d)
6 5
(a)
11.
If the area of a triangle ABC is a2 – (b – c)2, then tan A equals
(a)
- 8/15 (b)
15/16 (c)
8/15 (d)
15/8
12.
If cot −1
(
(a)
tan2 α
13.
x
2
36
(
)
cos α = u , then sin u equals
(b)
If cos −1 + cos −1
(a)
14.
)
cos α + tan −1
tan 2α
(c)
1
(d)
y
2
2
= θ . Then 9x – 12xy cos θ + 4y is equal to
3
(b)
-36 sin2 θ
(c)
36 sin2 θ
(d)
cot2 α/2
36 cos2 θ
If cos α + cos β = a, sin α + sin β = b then cos(α - β) is equal to
(a)
2ab
2
a + b2
(b)
a2 + b2
a 2 − b2
(c)
a2 + b2 − 2
2
11
(d)
(d)
b2 − a2
a2 + b2
15.
A
2
If cos A = ¾, then value of 32sin sin
(a)
16.
− 11
(c)
-11
π
−π
+θ ⋅
+ θ = k tan 3θ , then the value of k is
3
3
1
(b)
The value of cos
(a)
18.
(b)
If tan θ ⋅ tan
(a)
17.
11
5A
is
2
1
1/3
(c)
-1
(d)
none of these
2π
4π
6π
7π
+ cos
+ cos
+ cos
is
7
7
7
7
(b)
-1
(c)
½
If cos (θ + φ) = m cos (θ - φ), then tan θ =
(a)
[(1 + m)/(1 – m)] tan φ
(c)
[(1 – m)/(1 + m)] cot φ
(d)
-3/2
(b)
(d)
[(1 – m)/(1 + m)] tan φ
[(1 + m)/(1 – m)] sec φ
19.
If cos 20° = k and cos x = 2k2 – 1, then the possible values of x between 0°
and 360° are
(a)
140° (b)
40° and 140° (c)
40° and 320° (d)
50° and
130°
83
Punjab EDUSAT Society (PES)
MATHEMATICS
The equation a sin x + b cos x = c where c > a 2 + b 2 has
(a)
a unique solution
(b)
infinite number of
solutions
(c)
no solution
(d)
none of these
20.
21.
A tree is broken by wind, its upper part touches the ground at a point 10
metres from the foot of
the tree and makes an angles of 60° with the ground the
entire length of the tree is
(a)
22.
15 m (b)
20 m (c)
(
10 1 + 2
)
(d)
3
10 1 +
m
2
If the angles of triangle are 30° and 45° and the included side is
then the area of
(a)
(
)
3 + 1 cms,
triangle is
1
3 −1
3 +1
(b)
(c)
1
3 +1
(d)
none of these
23.
1
2
tan −1 + tan −1 =
11
12
33
tan −1
(b)
(a)
132
1
tan −1
2
(c)
132
tan −1
33
(d)
none of these
24.
If sin A = sin B and cos A = cos B, then A =
(a)
2nπ + B
(b)
2nπ - B
n
(d)
nπ + (-1) B
(c)
nπ + B
If the perimeter of a triangle ABC is 6 times the A.M.’s of the sine of its
25.
angles and the side a is 1,
then angle A is
(a)
30°
(b)
45°
(c)
60°
(d)
120°
26.
The general solution of the trigonometric equation sin x + cos x = 1 is given by (
n integer )
π
(a)
(b)
x = 2nπ +
x = 2nπ
2
(c)
27.
x = nπ + (−1)n
π
4
−
π
4
(d)
none of these
In a ∆ABC A : B : C = 3 : 5 : 4. Then a + b + c 2 is equal to
84
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
2b
(b)
2c
(c)
3b
(d)
3a
28.
In a triangle ABC, angle A is greater than angle B. If the measures of angles A
and B satisfy the
equation 3sin x – 4sin3 x – k = 0, 0 < k < 1, then the measure of
angle C is
(a)
π/3
(b)
π/2
(c)
2π/3 (d)
5π/6
π
3π
5π
7π
9π
11π
13π
29.
The value of sin sin sin sin sin sin
sin
is equal to
14
(a)
1/64
(b)
14
14
1/32
(c)
14
14
1/16
14
(d)
14
1/8
30.
The value of sin 10° + sin 20° + sin 30° + ………….. + sin 360° is
(a)
1
(b)
0
(c)
-1
(d)
none of these
31.
If x + 1/x = 2, the principle value of sin-1 x is
(a)
π/4
(b)
π/2
(c)
π
(d)
32.
33.
3π/2
The smallest angle of the triangle whose sides are 6 + 12, 48, 24 is
(a)
π/3
(b)
π/4
(c)
π/6
(d)
none of these
2
4
Given A = sin θ + cos θ , then for all real values of θ
(a)
1≤ A ≤ 2
(b)
3
≤ A ≤1
4
(c)
13
≤ A ≤1
16
(d)
3
13
≤ A≤
4
16
n
34.
Suppose sin 3 x sin 3x = ∑ Cm cos mx is an identity in x, where C0 , C1 ,.....Cn are
m =0
constants and
Cn ≠ 0. Then the value of n is
35.
(a)
6
(b)
3
The area of ∆ABC is
(a)
(c)
36.
37.
s ( s − a )( s − b )( s − c )
2
(c)
2
(d)
(b)
ab sin C
π
1 + cos
8
1
(a)
2
1
b 2 sin C sin A
sin B
1 a 2 sin B sin C
(d)
2
sin A
3π
5π
7π
1 + cos 1 + cos
1 + cos
is equal to
8
8
8
π
1
1+ 2
(b)
cos
(c)
(d)
8
8
2 2
The x satisfying sin-1 x + sin-1 (1 – x) = cos-1 x are
(a)
0
(b)
1, -1 (c)
0, ½
(d)
none of these
38.
If the radius of the circum circle of an isosceles triangle PQR is equal to PQ or
PR (PQ = PR)
Then the angle P must be
85
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
39.
π/6
(b)
π/3
Which is correct one?
(a)
sin 1° > sin 1 (b)
None of these
(c)
π/2
(d)
sin 1° = sin 1 (c)
2π/3
sin 1° = sin π/180 (d)
40.
The value of sin2 5° + sin2 10° + sin2 15° + ………. + sin2 90° is
(a)
9.5
(b)
9
(c)
10
(d)
8
41.
The value of the expression sin6 θ + cos6 θ + 3sin2 θcos2 θ
(a)
is 0
(b)
is 2
(c)
greater than 3
(d)
42.
The value of the expression cos 1° cos 2° ……….. cos 179° equals
(a)
43.
44.
0
(b)
1
1
2
(c)
(d)
If α + β = π/2 and β + γ = α, then tan α equals
(a)
2(tan β + tan γ)
(b)
tan β + tan γ
2tan β + tan γ
-1
(c)
tan β + 2tan γ (d)
The number of solutions of tan −1 x ( x + 1) + sin −1 x 2 + x + 1 =
(a)
45.
is -1
zero
y = sin −1
(a)
(b)
one
(c)
two
(d)
π
2
is
infinite
x
x
dy
+ cos −1 , then the value of
is :
2
2
dx
1
(b)
-1
(c)
0
(d)
2
46. On the bank of river there is a tree. On another bank, an observer makes an angle
of elevation of
60° at a top of the tree . The angle of elevation of the top of
the tree at a distance 20m away
from the bank is 30° .The width of the river is:
(a)
20m (b)
10m (c)
5m
(d)
1m
47.
If tan −1
(a)
48.
x −1 1
= tan −1 x , then value of x is:
x +1 2
1
1
(b)
(c)
3
2
3
(d)
2
Points D,E are taken on the side BC of the triangle ABC , such that
BD = DE = EC .
If ∠BAD = x ,
equal to:
(a)
1
∠DAE = y, ∠EAC = z , then the value of
(b)
2
(c)
4
(d)
sin( x + y )sin( y + z )
is
sin x sin z
none of these
86
Punjab EDUSAT Society (PES)
MATHEMATICS
49.
If p is the product of the sines of angles of a triangle and q be the product of
their cosines, then
tangents of the angle are roots of the equation :
3
(a)
qx − px 3 + (1 + q) x − p = 0
(b)
qx3 − qx 2 + (1 + p) x − q = 0
(c)
(1 + q ) x3 − px 2 + qx − p = 0
(d)
none of these
All the values of x for which expression (1 + tan x + tan 2 x) (1 − cot x + cot 2 x) is
50.
positive satisfy
π
(a)
0≤ x≤
(b)
(c) for all x ∈ R (d)
0≤ x ≤π
x=0
2
51.
3
has:
2
The equation sin −1 x − cos −1 x = cos−1
(a)
(c)
no solution
(b)
infinite number of solutions (d)
unique solution
none of these
52.
The angle of a elevation of the top of the tower observed from each of the
three points A, B,
Cone the ground forming a triangle is the same angle α. If R is
ABC , then the height of the tower is:
the circum-radius of the triangle
(a)
(b)
(c) R cot α (d)
R sin α
R cos α
R tan α
53.
In a triangle ∆ABC , a , c, A are given and b1 , b2 are two values, of the third side
b is, such that b2 = 2b1 then sin A is equal to :
(a)
9a 2 − c 2
8a 2
(b)
9a 2 − c 2
8a 2
(c)
9a 2 + c 2
8a 2
(d)
none of
these
54.
The angle of elevation of top of a tower from a point on the ground is 30° and
it is 60° when it is
viewed from a point located 40 m away from the initial point
towards the tower. The height of
the tower is
(a)
(c)
−
3
m
20
(d)
−20 3m
(b)
3
m
20
20 3m
55.
In ∆ABC, ∠A = π/2, b = 4, c = 3, then the value of R/r is equal to
(a)
5/2
(b)
7/2
(c)
9/2
(d)
35/24
56.
If a2, b2, c2 are in A.P, then which of the following is also in A.P?
(a)
sin A, sin B, sin C
(b)
tan A, tan B, tan C
(c)
cot A, cot B, cot C
(d)
none of these
The solution set of the equation sin-1 x = 2tan-1 x is:
(a)
{1, 2} (b)
{-1, 2} (c)
{-1, 1, 0}
(d)
{1, ½, 0}
57.
58.
Let cos(2tan-1 x) = ½, then the value of x is
87
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
59.
If sin −1
(a)
60.
(b)
3
1
3
(c)
1− 3
(d) 1 −
1
3
2
2a
2x
−1 1 − b
−
cos
= tan −1
, then the value of x is
2
2
1+ a
1+ b
1 − x2
a+b
a −b
a
(b)
b
(c)
(d)
1 − ab
1 + ab
θ +α θ −α
If cos θ = cos α cos β, then tan
tan
is equal to
2 2
tan 2
(a)
cot 2
β
α
(b)
2
tan 2
β
(c)
2
tan 2
θ
(d)
2
2
61.
A house subtends a right angle at the window of an opposite house and the
angle of elevation
of the window from the bottom of the first house is 60° . If the
distance between the two
houses be 6m, then the height of the first house is
62.
(a)
8 3m
(b)
6 3m
(c)
4 3m
None of these
8
If cos θ = and θ lies in the Ist quadrant, then the value of
17
cos(30° + θ ) + cos(45° − θ ) + cos(120° − θ )
(a)
(c)
63.
23 3 − 1 1
+
17 2
2
23 3 − 1 1
−
17 2
2
(b)
(d)
The root of the equation 1 − cos θ = sin θ .sin
(a)
kπ , k ∈ I
(b)
2kπ , k ∈ I
(d)
23 3 + 1 1
+
17 2
2
23 3 + 1 1
−
17 2
2
θ
is
2
(c)
k
π
,k ∈I
2
(d)
None of these.
64.
In ∆ABC , if sin 2
(a)
65.
AP
A
C
, sin 2 be in H.P. then a, b, c will be in
2
2
(b)
GP
(c)
HP
A
B
− tan
2 is equal to
In any triangle ABC , 2
A
B
tan + tan
2
2
(d)
None of these
88
Punjab EDUSAT Society (PES)
MATHEMATICS
a−b
a+b
(a)
(b)
a −b
c
(c)
a −b
a+b+c
(d)
c
a+b
66.
The sides of triangle are 3 x + 4 y,4 x + 3 y and 5 x + 5 y where x , y > 0 then the
triangle is
(a)
right angle
(b)
obtuse angled (c)
equilateral
(d)
none of these
In a triangle with sides a, b, c, r1 > r2 > r3 (which are the ex-radii) then
(a)
(b)
(c) a > b and b < c
(d)
a>b>c
a<b<c
67.
a < b and b > c
68.
The sum of the radii of inscribed and circumscribed circles for an n sided
regular polygon of side a , is
a
a
π
π
π
(a)
cot
(b)
a cot
(c)
cot
(d)
2n
4
n
2
2n
π
a cot
2n
69.
In a triangle ABC, medians AD and BE are drawn. If AD = 4, ∠DAB = π/6
and ∠ABE = π/3, then
the area of the ∆ABC is
(a)
64/3
(b)
If in a triangle ABC a cos 2
70.
(a)
satisfy a + b = c
are in H.P
8/3
(c)
16/3
32
3/ 3
(d)
C
A 3b
+ c cos 2 =
, then the sides a, b and c
2
2 2
(b)
are in A.P
(c)
are in G.P
(d)
If π < α - β < 3π, sin α + sin β = - 21/65 and cos α + cos β = - 27/65, then
α −β
cos
is
71.
2
(a)
- 6/65 (b)
3
(c)
130
6/65
(d)
−
3
130
72.
If u = a 2 cos 2 θ + b 2 sin 2 θ + a 2 sin 2 θ + b 2 cos 2 θ , then the difference between
the maximum and minimum value of u2 is given by
(a)
(b)
(c)
(d)
2 a 2 + b2
( a − b )2
( a + b )2
(
2 a 2 + b2
)
89
Punjab EDUSAT Society (PES)
MATHEMATICS
The sides of a triangle are sin α, cos α and 1 + sin α cos α for some 0 < α <
73.
π/2. Then the greatest angle of the triangle is
(a)
150° (b)
90°
(c)
120° (d)
60°
74.
A person standing on the bank of river observes that the angle of elevation of
the top of a tree
on the opposite bank of the river is 60° and when he retires 40
meters away from the tree the
angle of elevation becomes 30°. The breath of
the river is
(a)
60 m (b)
30 m (c)
40 m (d)
20 m
If cos-1 x – cos-1 y/2 = α, then 4x2 – 4xy cos α + y2 is equal to
(a)
2sin 2α
(b)
4
(c)
4sin2 α
75.
(d)
-
4sin α
76.
If in a ∆ABC, the altitudes from the vertices A, B, C on opposite sides are in
H.P., then
sin A. Sin B. sin C are in
(a)
G.P
(b)
A.P (c)
A.G.P
(d)
H.P.
2
77.
In triangle PQR, ∠R = π/2. If tan P/2 and tan Q/2 are roots of the equations
ax2 + bx + c = 0 (a≠0) then
(a)
a+b=c
(b)
b+c=a
(c)
a+c=b
(d)
b=
c
78.
In a triangle ABC, let ∠C =
π
2
. If r is the inradius and R is the circumradius of
triangle, then
2(r + R) is equal to
(a)
a + b (b)
b + c (c)
c + a (d)
79.
If 0 < x < π, and cos x + sin x = ½, then tan x is
(a)
(c)
− 4 + 7 / 3 (b)
1+ 7 / 4
(
(4 − 7 ) / 3
)
(
)
a+b–c
(1 − 7 ) / 4
(d)
80.
The number of values of x in the interval [0, 3π] satisfying the equation 2sin2
x + 5sin x – 3 = 0 is
(a)
1
(b)
2
(c)
3
(d)
6
x
5 π
If sin −1 + cos ec −1 = then a value of x is
5
4 2
(a)
3
(b)
4
(c)
5
(d)
1
82.
A tower stands at the centre of a circular park .A and B are two points on the
boundary of the park such that AB (= a ) subtends an angle of 60° at the
foot of the tower , and the
angle of elevation of the top of the tower from A or B
81.
90
Punjab EDUSAT Society (PES)
MATHEMATICS
is 30° . The height of the tower is
(a)
83.
2a 3
(b)
a/ 3
5
2
The value of cot cos ec −1 + tan −1 is
3
3
(c)
a 3
(d)
2a / 3
4
5
6
3
(b)
(c)
(d)
17
17
17
17
84.
AB is a vertical pole with B at the ground level and A at the top. A man finds
that the angle of
elevation of the point A from a certain point C on the ground is
60° . He moves away from the
pole along the line BC to a point D such that CD
= 7m. From D the angle of elevation of the point
A is 45°. Then the height of the
pole is
7 3
7 3 1
7 3 1
(a)
( 3 − 1)m (b)
m (c)
m (d)
2
2
2
3 +1
3 −1
(a)
7 3
( 3 + 1)m
2
1.
7.
13.
19.
25.
c
d
c
c
a
2.
8.
14.
20.
26.
d
b
c
c
c
Answers
3.
d
9.
c
15.
c
21.
c
27.
c
4.
10.
16.
22.
28.
c
c
c
a
c
5.
11.
17.
23.
29.
c
c
d
d
a
6.
12.
18.
24.
30.
b
c
c
c
b
31.
37.
43.
b
c
c
32.
38.
44.
c
d
c
33.
39.
45.
b
c
c
34.
40.
46.
a
a
b
35.
41.
47.
d
d
b
36.
42.
48.
c
a
c
49.
a
50.
c
51.
b
52.
d
53.
b
54.
c
55.
a
56
c
57.
c
58
b
59
d
60
b
61.
66.
a
b
62.
67.
a
a
63.
68.
b
c
64.
69.
c
d
65.
70.
b
b
71.
d
72.
a
73.
c
74.
d
75.
c
76.
b
77.
a
78.
84.
a
d
79.
a
80.
c
81.
a
82.
b
83.
c
91
Punjab EDUSAT Society (PES)
MATHEMATICS
PROBLEM SET ON VECTORS & 3D
1.
A particle acted on by constant forces 4iˆ + ˆj − 3kˆ and 3iˆ + ˆj − kˆ to the point
5iˆ + 4 ˆj − kˆ . The
(a)
(d)*
total work done by the forces is
50 units
(b)
20 units
40 units
(c)
30 units
r
r
The vectors AB = 3iˆ + 4kˆ and AC = 5iˆ − 2 ˆj + 4kˆ are the sides of a triangle ABC.
2.
The length of the median through A is
(a)
288
(b)
18
(c)
72
(d)*
33
3.
Consider points A, B, C and D with position vectors
7iˆ − 4 ˆj + 7 kˆ, iˆ − 6 ˆj + 10kˆ, − iˆ − 3 ˆj + 4 kˆ
and 5iˆ − ˆj + 5kˆ respectively. Then quadrilateral ABCD is a
4.
(a)* Parallelogram but not a rhombus
(b)
Square
(c)
Rhombus
(d)
Rectangle
r r
r
Let a , b and c be three non-zero vectors such that no two of these are
r r
r
r
r
r
a + 2b collinear with c and b + 3c is collinear with a (λ
r r
r
being some non-zero scalar) then
a + 2b + 6c equals
r
r
r
(a)
0
(b)
λb
(c)*
λc
(d)
λa
5.
A particle is acted upon by constant forces 4iˆ + ˆj − 3kˆ and 3iˆ + ˆj − kˆ which
collinear. If the vector
displace it from a
point iˆ + 2 ˆj + 3kˆ to the point 5iˆ + 4 ˆj + kˆ . The work done in
standard units by the forces is
given by
(a)
15
(b)
30
(c)
25
(d)* 40
6.
7.
r r r
If a , b , c are non-coplanar, then the vectors
are non-coplanar for
(a)
no value of λ
(c)* all except two values of λ
r r r
r
r
r
Let u , v , w be such that u = 1, v = 2, w = 3 .
r
to that of w
r
r r
r r
r
r
a + b + 3c , λ b + 4c and ( 2λ − 1) c
(b)
all except one value of λ
(d)
all values of λ
r
r
If the projection v along u is equal
r r
r
r
r
along u and v , w are perpendicular to each other then u − v + w equals
(a)
14
(b)
7
(c)*
14
(d)
2
r
r
r
r
r
r
r 1 r r
8.
Let a , b and c be non-zero vectors such that a × b × c = b c a . If θ is the
3
r
r
acute angle between the vectors b and c , then sin θ equals
(
(a)*
9.
2 2
3
(b)
2
3
(c)
2
3
)
(d)
1
3
If C is the mid point of AB and P is any point outside AB, then
r r
r r
r
r
(a)*
PA + PB = 2 PC
(b)
PA + PB = PC
r r
r
r
r
r
r r
(c)
PA + PB + 2 PC = 0
(d)
PA + PB + PC = 0
92
Punjab EDUSAT Society (PES)
MATHEMATICS
10.
(r )
For any vector a , the value of ( a × iˆ ) + ( a × ˆj ) + a × kˆ is equal to
r
r
r
2
r
r
2
2
r
r
(a)
3a 2
(b)
a2
(c)*
2a 2
(d)
4a 2
11.
Let a, b and c be distinct non-negative numbers. If the vectors
ˆ
ˆ
ai + aj + ckˆ, iˆ + kˆ and
ciˆ + cjˆ + bkˆ lie in a plane, then c is
(a)* the Geometric Mean of a and b
Mean of a and b
(c)
equal to zero
and b
12.
13.
r r r
r
λ ar + b
(b)
(d)
the
arithmetic
the Harmonic mean of a
If a , b , c are non coplanar vectors and λ is a real number then
(
)
r
r
r
r
r
r
(a)
(c)
λ 2b λ c = a b + c b for
exactly one value of λ
exactly three values of λ
Let
r
r
r
a = iˆ − kˆ, b = xiˆ + yjˆ + (1 − x ) kˆ, c = yiˆ + xjˆ + (1 + x − y ) kˆ ,
depends upon
(a)
only x
both x and y
(b)
only y
(b)*
(d)
(c)*
no value of λ
exactly two values of λ
then
r r r
a, b , c
neither x or y (d)
14.
The two lines x = ay + b, z = cy + d; and x = a′y + b′, z = c′y + d′ are
perpendicular to each other if
(a)
a c
+ = −1
a′ c′
(b)
a c
+ =1
a ′ c′
(c)*
aa′ + cc′ = -1 (d)
aa′
+ cc′ = 1
15.
The values of a, for which the points A, B, C with position vectors
2iˆ − ˆj + kˆ, iˆ − 3 ˆj − 5kˆ and aiˆ − 3 ˆj + kˆ respectively are the vertices of a rightangled triangle with C = π/2 are
(a)
-2 and 1
(b)
2 and -1
(c)* 2 and 1
(d)
-2
and -1
16.
If û and v̂ are unit vectors and θ is the acute angle between them , then
2uˆ × 2vˆ is a unit vec tor for
(b)
No value of θ
(a)
More than two values of θ
(c)* exactly one value of θ
(d)
exactly two values of θ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
17.
Let a = i + j + k , b = i − j + 2 k and c = xi + ( x − 2) j − kˆ. If the vector c lies in
the
plane of a and b , then x equals
(a)
1
(b)
-4
(c)* - 2
(d)
0
18.
Let L be the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2z
= 2. If L makes an angle α with the positive x-axis, then cos α equals
93
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
19.
½
(b)
1
If a line makes an angle of
(c)
1
2
(d)*
1
3
π
with the positive directions of each of x-axis
4
and
y-axis, then the angle that the line makes with the positive direction of the z-axis
is
(a)
π/3
(b)
π/4
(c)* π/2
(d)
π/6
20.
If û and v̂ are unit vectors and θ is the acute angle between them , then
2uˆ × 2vˆ is a unit
vector for
(a)
More than two values of θ (b)
No value of θ
(c)* exactly one value of θ
(d)
exactly two values of θ
21.
If (2,3,5) is one end of a diameter of the sphere
x 2 + y 2 + z 2 − 6 x − 12 y − 2 z + 20 = 0, then the coordinates of the other end
of the diameter are
(a)
(4 , -3 , 3 ) (b) ( 4 , 3 , 5 ) (c) ( 4 , 3 , -3 ) (d)*
( 4 , 9 , -3 )
ˆ
ˆ
ˆ
22.
Let a = iˆ + ˆj + k , b = iˆ − ˆj + 2 k and c = xiˆ + ( x − 2) ˆj − k . If the vector c lies in
the
plane of a and b , then x equals
(a)
1
(b)
-4
(c)* - 2
(d)
0
r
r
ˆ
ˆ
ˆ
23.
The vector a = α i + 2 j + βk lies in the plane of the vectors b = iˆ + ˆj and
r
r
r
c = ˆj + kˆ bisects the angle between b and c . Then which one of the following
gives possible values of α and β?
(a)
α = 2, β = 1
(b)* α = 1, β = 1
(c)
α = 2, β = 2
(d)
α = 1, β = 2
r
r
r r
r
24.
The non-zero vectors a , b and c are related by and c = −7b . then the angle
r
r
between a and
c is
π
π
(a)
(b)* π
(c)
0
(d)
2
4
25.
The line passing through the points (5,1, a ) and (3, b,1) crosses the yz-plane
17 −13
at the point
0, ,
. Then
2 2
(a)*
a = 6, b = 4 (b)
a = 8, b = 2 (c)
a = 2 , b = 8 (d)
a = 4,b = 6
x −1 y − 2 z − 3
x − 2 y − 3 z −1
26.
If the straight lines
=
=
and
=
=
intersect at a
k
2
3
3
k
2
point, then
the integer k is equal to
(a)
2
(b)
−2
(c)* -5
(d)
5
27.
r r r
r r r
r r
r r
u = a − b and v = a + b and a = b = 2 , then ( u × v ) is equal to
94
Punjab EDUSAT Society (PES)
MATHEMATICS
(
r r
2 16 − a ⋅ b
(a)*
(
r r
4 − a ⋅b
28.
29.
then
r
r
)
)
2
(
r r
2 4 − a ⋅b
(b)
)
2
( )
r r
16 − a ⋅ b
(c)
2
(d)
2
r
r
r
r r
r
r r
r r
If a + b + c = 0, and a = 2, b = 3, c = 5 then a ⋅ b + b ⋅ c + c ⋅ a =
(a)
19
(b)* -19
(c)
61
(d)
-61
r
r
In ∆ABC, AB = riˆ + ˆj , AC = siˆ − ˆj . If the area of triangle is of unit magnitude,
r−s =2
(a)
r + s =1
(b)
(c)*
r+s =2
(d)
r − s =1
aˆ , bˆ, cˆ are three unit vectors, â and bˆ are perpendicular to each other and vector
ĉ is equally inclined to both â and bˆ at an angle θ. If cˆ = α aˆ + β bˆ + γ aˆ × bˆ , where
30.
(
)
α, β, γ are constants then
(a)
α = β = -cos θ, γ2 = cos 2θ
(b)
α = β = cos θ, γ2 = cos 2θ
(c)* α = β = cos θ, γ2 = -cos 2θ
(d)
α = β = -cos θ, γ2 = -cos
2θ
r
r r r r
r r
r
r
31.
If a makes an acute angle with b and r × b = c × b and r ⋅ a = 0, then r is equal
to
r r r
a×c −b
(a)
r r
c×a
(b)
(c)*
r r
r c ⋅a r
c+ r rb
b ⋅a
32.
r
r
r
r r
r r
(b)
(
r r
r r
6 c ×b
r r
r c ⋅a r
c− r rb
b ⋅a
If a + 2b + 3c = 0 and a × b + b × c + c × a is equal to
r r
3( a × c )
(a)
(
r r
2 b ×a
)
)
(c)*
(
r r
2 a ×b
(d)
)
(d)
33.
Three consecutive vertices of a rhombus have the position vectors
iˆ + 2 ˆj + 3kˆ, 2iˆ + 4 ˆj + 5kˆ and
4iˆ + 5 ˆj + 3kˆ, then the position vector of fourth vertex is
3iˆ + 3 ˆj + 3kˆ
3iˆ + 3 ˆj + kˆ
(a)
r
r
r
r r
3iˆ − 3 ˆj + 3kˆ
(b)
r
r
r
(c)
−3iˆ + 3 ˆj + 3kˆ
(d)*
r r r
If a + b + c = α d , b + c + d = β a and a , b , c are non-coplanar, then the sum
r
r r
a + b + c + d equals
r
r
r
r
r
(a)*
0
(b)
(c)
( β − 1) d + (α − 1) a
(α − 1) d + ( β − 1) a
34.
r
35.
r
(d)
(α − 1) d − ( β − 1) a
The volume of a tetrahedron, the position vectors of whose vertices are
5iˆ − ˆj + kˆ, 7iˆ − 4 ˆj + 7 kˆ, iˆ − 6 ˆj + 10 kˆ and −iˆ − 3 ˆj + 7 kˆ , is
95
Punjab EDUSAT Society (PES)
MATHEMATICS
(a)
15
(b)
r r
3
(c)*
11
(d)
7
r
36.
Let a , b and c be three non-zero vectors, no two of which are collinear and the
r r
r
r r
r r r
r
vectors a + b is
collinear with c while b + c is collinear with a . Then a + b + c =
r
r
r
(b)
b
(c)
(d)* none of
(a)
a
c
these
r r r
37.
If a , b , c are three non-zero vectors such that
r r r
r r r r r r
a + b + c = 0 and m = a ⋅ b + b ⋅ c + c ⋅ a , then
(a)* m < 0
(b)
m>0
(c)
1
m=0
(d)
m=
r
r
r r
r
r
38.
The vector a = xiˆ + yjˆ + zkˆ, b = ˆj and c are such that a , b , c from a right handed
r
system, then vector c is
(a)
0
(b)
yjˆ
(c)*
ziˆ − xkˆ
(d)
− ziˆ + xkˆ
r r
39.
r
If a , b and c be vectors with magnitudes 3, 4 and 5 respectively and
r r r
r r r r r r
a + b + c = 0 , then the value of a ⋅ b + b ⋅ c + c ⋅ a is
(a)
47
(b)
25
(c)
50
(d)* -25
r
r
r ˆ ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
40.
Let a = 2i − j + k , b = i + 2 j − k and c = i + j − 2 k be three vectors. A vector in
the plane of
(a)*
r
r
r
b and c whose projection on a is magnitude 2 / 3 is
(b)
(c)
2iˆ + 3 ˆj − 3kˆ
2iˆ + 3 ˆj + 3kˆ
−2iˆ + 5 ˆj + 5kˆ
(d)
2iˆ + ˆj + 5kˆ
r r
r r r
r r r b +c
41.
If a , b , c are non-coplanar unit vectors such that a × b × c =
, then the
2
r
r
angle between a and c is
(
(a)*
3π/4
(b)
π/4
)
π/2
(c)
(d)
π
A unit vector in xy -plane that makes an angle of 45° with the vector iˆ + ˆj and
42.
an angle of 60°
with the vector 3iˆ − 4 ˆj is
iˆ
(a)
(b)
iˆ + ˆj
2
(c)
ur ur ur
ur
iˆ − ˆj
2
(d)* none of
these
r
43.
ur ur
ur
r
0
(b)
ur ur
ur ur
If A × ( B × C ) = B × (C × A) and [ A B C ] ≠ 0 then A × ( B × C ) is equal to
(a)*
ur ur
A× B
(c)
ur ur
B×C
(d)
ur ur
C×A
r r ur
44.
If u , v , w are three non coplanar vectors , then
r r ur r r r ur
(u + v − w ).((u − v) × (v − w)) equals
96
Punjab EDUSAT Society (PES)
MATHEMATICS
r r ur
u.(v × w)
(a)*
0
r
r r ur
u .v × w
(b)
r r ur
3u . v × w
(c)
(d)
r
ur
45.
Let a = 2iˆ + ˆj + kˆ , b = iˆ + 2 ˆj − kˆ and c be a coplanar unit vector perpendicular to
r
r
a then c =
(a)*
− ˆj + kˆ
(b)
−iˆ − ˆj − kˆ
iˆ − 2 ˆj
5
(c)
iˆ − ˆj − kˆ
(d)
2
3
3
r
r
r
ur
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
46.
If a = i − j , b = i + j , c = i + 3 j + 5k and n be a unit vector such that
rr
rr
rr
b .n = 0, a .n = 0, then The
the value of c .n is
47.
(a)
1
(b)
3
(c)* 5
(d)
2
r
r
r
r r r
r r r r r r
If a = 3, b = 5 and c = 4 and a + b + c = 0, then the value of a .b + b . c + c .a is
equal to :
(a)
0
(b)*
-25
(c)
25
(d)
48.
The magnitude of cross product of two vectors is
The angle between the vectors is:
π
π
π
(a)
(b)*
(c)
6
3
none of these
3 times the dot product.
2
π
(d)
4
49.
The summation of two unit vectors is a third unit vector, then the modulus of
the difference of
the unit vectors is
(a)*
50.
(
3
(b)
1− 3
(c)
1 + 3 (d)
− 3
2iˆ − 3 ˆj − 6kˆ
7
If M denotes the mid point of the line joining
)
(
)
A 4iˆ + 5 ˆj − 10kˆ and B −iˆ + 2 ˆj + kˆ , then
equation of the plane through M and
perpendicular to AB is
(a)*
(c)
(
(
)
r
135
r −5iˆ − 3 ˆj + 11kˆ +
=0
2
r
r 4iˆ + 5 ˆj − 10kˆ + 4 = 0
)
(b)
(d)
r 3
7
9 135
r iˆ + ˆj − kˆ +
=0
2
2 2
2
r
r −iˆ + 2 ˆj + kˆ + 4 = 0
(
)
97
Punjab EDUSAT Society (PES)
Related documents